🧠:<think>Alright, so I have this problem about designing an energy storage system for a wind farm. It involves both a flywheel and lithium-ion batteries. The first part is about deriving the angular velocity of the flywheel as a function of time, and the second part is about calculating the battery storage capacity. Let me try to tackle each part step by step.Starting with part 1: The flywheel is a solid cylinder with mass ( m ) and radius ( r ). The wind farm generates power ( P(t) = P_0 sin(omega t) ). I need to find the angular velocity ( omega_f(t) ) of the flywheel, assuming it starts from rest and all the generated power is initially stored in the flywheel.Okay, so power is the rate of doing work, which is also the rate of energy transfer. So, the power ( P(t) ) is equal to the time derivative of the kinetic energy stored in the flywheel. The kinetic energy ( K ) of a rotating object is given by ( K = frac{1}{2} I omega_f^2 ), where ( I ) is the moment of inertia.For a solid cylinder, the moment of inertia ( I ) is ( frac{1}{2} m r^2 ). So, substituting that into the kinetic energy formula, we get ( K = frac{1}{2} times frac{1}{2} m r^2 times omega_f^2 = frac{1}{4} m r^2 omega_f^2 ).Now, power ( P(t) ) is the derivative of kinetic energy with respect to time, so:( P(t) = frac{dK}{dt} = frac{d}{dt} left( frac{1}{4} m r^2 omega_f^2 right) )Taking the derivative, we get:( P(t) = frac{1}{4} m r^2 times 2 omega_f times frac{domega_f}{dt} = frac{1}{2} m r^2 omega_f alpha )Where ( alpha = frac{domega_f}{dt} ) is the angular acceleration.But wait, power can also be expressed as torque multiplied by angular velocity, ( P = tau omega_f ). Torque ( tau ) is related to angular acceleration by ( tau = I alpha ). So, substituting torque into the power equation:( P(t) = I alpha omega_f )Which is the same as the previous expression since ( I = frac{1}{2} m r^2 ). So, that's consistent.So, from ( P(t) = I alpha omega_f ), we can write:( alpha = frac{P(t)}{I omega_f} )But ( alpha = frac{domega_f}{dt} ), so:( frac{domega_f}{dt} = frac{P(t)}{I omega_f} )Hmm, this gives a differential equation:( omega_f frac{domega_f}{dt} = frac{P(t)}{I} )Which is a separable equation. Let me write it as:( omega_f domega_f = frac{P(t)}{I} dt )Integrating both sides from time 0 to time ( t ), with ( omega_f(0) = 0 ):( int_0^{omega_f(t)} omega_f domega_f = int_0^t frac{P(t')}{I} dt' )Calculating the left integral:( frac{1}{2} omega_f(t)^2 = frac{1}{I} int_0^t P(t') dt' )So,( omega_f(t)^2 = frac{2}{I} int_0^t P(t') dt' )Taking the square root:( omega_f(t) = sqrt{frac{2}{I} int_0^t P(t') dt'} )Substituting ( I = frac{1}{2} m r^2 ):( omega_f(t) = sqrt{frac{2}{frac{1}{2} m r^2} int_0^t P(t') dt'} = sqrt{frac{4}{m r^2} int_0^t P(t') dt'} )Simplify the constants:( omega_f(t) = frac{2}{r sqrt{m}} sqrt{ int_0^t P(t') dt' } )But ( P(t) = P_0 sin(omega t) ), so let's substitute that in:( int_0^t P(t') dt' = int_0^t P_0 sin(omega t') dt' = frac{P_0}{omega} (1 - cos(omega t)) )So, substituting back:( omega_f(t) = frac{2}{r sqrt{m}} sqrt{ frac{P_0}{omega} (1 - cos(omega t)) } )Simplify the square root:( sqrt{1 - cos(omega t)} = sqrt{2 sin^2left( frac{omega t}{2} right)} = sqrt{2} sinleft( frac{omega t}{2} right) )So,( omega_f(t) = frac{2}{r sqrt{m}} sqrt{ frac{P_0}{omega} } sqrt{2} sinleft( frac{omega t}{2} right) )Simplify constants:( frac{2}{r sqrt{m}} times sqrt{2} = frac{2 sqrt{2}}{r sqrt{m}} )So,( omega_f(t) = frac{2 sqrt{2}}{r sqrt{m}} sqrt{ frac{P_0}{omega} } sinleft( frac{omega t}{2} right) )Let me combine the constants:( sqrt{ frac{P_0}{omega} } times frac{2 sqrt{2}}{r sqrt{m}} = frac{2 sqrt{2 P_0}}{sqrt{omega} r sqrt{m}} )Which can be written as:( frac{2 sqrt{2 P_0}}{r sqrt{m omega}} )So, the expression becomes:( omega_f(t) = frac{2 sqrt{2 P_0}}{r sqrt{m omega}} sinleft( frac{omega t}{2} right) )Hmm, let me double-check the steps.1. Expressed power as derivative of kinetic energy.2. Used the kinetic energy of a solid cylinder.3. Derived the differential equation.4. Separated variables and integrated.5. Substituted the integral of ( P(t) ).6. Simplified using trigonometric identity.Seems okay. Maybe I can write it as:( omega_f(t) = sqrt{frac{2}{I} int_0^t P(t') dt'} )But substituting ( I ) and ( P(t) ) gives the expression above. Alternatively, perhaps factor differently.Alternatively, let's think about units to check.Power has units of Watts, which is J/s or kg m²/s³.Moment of inertia ( I ) has units kg m².So, ( sqrt{frac{2}{I} times text{Energy}} ) would be ( sqrt{frac{2}{kg m²} times kg m²/s²} = sqrt{1/s²} = 1/s ), which is correct for angular velocity.So, units check out.Alternatively, perhaps express the integral as:( int_0^t P(t') dt' = frac{P_0}{omega} (1 - cos(omega t)) )So, plugging back into ( omega_f(t) ):( omega_f(t) = sqrt{frac{2}{I} times frac{P_0}{omega} (1 - cos(omega t))} )Which is another way to write it, perhaps more compact.So, maybe that's the expression they want.Alternatively, since ( 1 - cos(omega t) = 2 sin^2(omega t / 2) ), so:( omega_f(t) = sqrt{frac{2}{I} times frac{P_0}{omega} times 2 sin^2(omega t / 2)} = sqrt{frac{4 P_0}{I omega} sin^2(omega t / 2)} = frac{2}{sqrt{I omega}} sqrt{P_0} sin(omega t / 2) )Which is similar to what I had before.Since ( I = frac{1}{2} m r^2 ), so ( sqrt{I} = sqrt{frac{1}{2} m r^2} = r sqrt{frac{m}{2}} ). So,( frac{2}{sqrt{I omega}} = frac{2}{r sqrt{frac{m}{2} omega}} = frac{2}{r} times sqrt{frac{2}{m omega}} = frac{2 sqrt{2}}{r sqrt{m omega}} )Which is the same as before.So, the expression is consistent.Therefore, the angular velocity as a function of time is:( omega_f(t) = frac{2 sqrt{2 P_0}}{r sqrt{m omega}} sinleft( frac{omega t}{2} right) )Alternatively, written as:( omega_f(t) = sqrt{frac{4 P_0}{m r^2 omega}} sinleft( frac{omega t}{2} right) )Either form is acceptable, I think.Moving on to part 2: The lithium-ion batteries take over once the flywheel reaches its maximum safe angular velocity ( omega_{text{max}} ). The total energy generated over a period ( T ) is ( E_{text{total}} = int_0^T P(t) dt ). The flywheel reaches ( omega_{text{max}} ) at time ( t_1 ), and the batteries store the remaining energy from ( t_1 ) to ( T ). We need to calculate the required storage capacity ( C ) of the batteries.So, first, we need to find ( t_1 ) when ( omega_f(t_1) = omega_{text{max}} ).From part 1, we have ( omega_f(t) = frac{2 sqrt{2 P_0}}{r sqrt{m omega}} sinleft( frac{omega t}{2} right) ). Setting this equal to ( omega_{text{max}} ):( omega_{text{max}} = frac{2 sqrt{2 P_0}}{r sqrt{m omega}} sinleft( frac{omega t_1}{2} right) )Solving for ( sinleft( frac{omega t_1}{2} right) ):( sinleft( frac{omega t_1}{2} right) = frac{omega_{text{max}} r sqrt{m omega}}{2 sqrt{2 P_0}} )Let me denote ( sin(theta) = frac{omega_{text{max}} r sqrt{m omega}}{2 sqrt{2 P_0}} ), where ( theta = frac{omega t_1}{2} ).So, ( theta = arcsinleft( frac{omega_{text{max}} r sqrt{m omega}}{2 sqrt{2 P_0}} right) )Therefore,( t_1 = frac{2}{omega} arcsinleft( frac{omega_{text{max}} r sqrt{m omega}}{2 sqrt{2 P_0}} right) )Assuming that the argument inside arcsin is less than or equal to 1, which it should be since ( omega_{text{max}} ) is the maximum safe angular velocity.Now, the total energy generated over period ( T ) is ( E_{text{total}} = int_0^T P(t) dt = int_0^T P_0 sin(omega t) dt )Calculating this integral:( E_{text{total}} = frac{P_0}{omega} (1 - cos(omega T)) )But we also need to find the energy stored in the flywheel up to time ( t_1 ), which is ( E_{text{flywheel}} = int_0^{t_1} P(t) dt = frac{P_0}{omega} (1 - cos(omega t_1)) )Therefore, the energy stored in the batteries is the remaining energy:( E_{text{batteries}} = E_{text{total}} - E_{text{flywheel}} = frac{P_0}{omega} [ (1 - cos(omega T)) - (1 - cos(omega t_1)) ] = frac{P_0}{omega} ( cos(omega t_1) - cos(omega T) ) )But wait, actually, the flywheel stops storing energy at ( t_1 ), so from ( t_1 ) to ( T ), the batteries store the energy. So, the energy stored in batteries is ( int_{t_1}^T P(t) dt ).Which is:( E_{text{batteries}} = int_{t_1}^T P(t) dt = frac{P_0}{omega} ( cos(omega t_1) - cos(omega T) ) )So, the required storage capacity ( C ) of the batteries is this energy.But let me express ( cos(omega t_1) ) in terms of ( omega_{text{max}} ).From part 1, we have:( sinleft( frac{omega t_1}{2} right) = frac{omega_{text{max}} r sqrt{m omega}}{2 sqrt{2 P_0}} )Let me denote ( sin(theta) = k ), where ( theta = frac{omega t_1}{2} ), so ( cos(2theta) = 1 - 2 sin^2(theta) ).Therefore,( cos(omega t_1) = cos(2theta) = 1 - 2 k^2 = 1 - 2 left( frac{omega_{text{max}}^2 r^2 m omega}{8 P_0} right ) = 1 - frac{omega_{text{max}}^2 r^2 m omega}{4 P_0} )So,( E_{text{batteries}} = frac{P_0}{omega} left( 1 - frac{omega_{text{max}}^2 r^2 m omega}{4 P_0} - cos(omega T) right ) )Simplify:( E_{text{batteries}} = frac{P_0}{omega} - frac{omega_{text{max}}^2 r^2 m}{4} - frac{P_0}{omega} cos(omega T) )But wait, ( frac{P_0}{omega} (1 - cos(omega T)) ) is the total energy, and ( E_{text{flywheel}} = frac{P_0}{omega} (1 - cos(omega t_1)) ), so subtracting gives ( E_{text{batteries}} = frac{P_0}{omega} ( cos(omega t_1) - cos(omega T) ) ), which is the same as above.Alternatively, using the expression for ( cos(omega t_1) ):( E_{text{batteries}} = frac{P_0}{omega} left( 1 - frac{omega_{text{max}}^2 r^2 m omega}{4 P_0} - cos(omega T) right ) )But perhaps it's better to express it in terms of ( cos(omega t_1) ) as we did earlier.Alternatively, let's think about the energy stored in the flywheel at ( t_1 ):( E_{text{flywheel}} = frac{1}{2} I omega_{text{max}}^2 = frac{1}{2} times frac{1}{2} m r^2 times omega_{text{max}}^2 = frac{1}{4} m r^2 omega_{text{max}}^2 )But also, ( E_{text{flywheel}} = int_0^{t_1} P(t) dt = frac{P_0}{omega} (1 - cos(omega t_1)) )So, equating these:( frac{1}{4} m r^2 omega_{text{max}}^2 = frac{P_0}{omega} (1 - cos(omega t_1)) )Therefore,( 1 - cos(omega t_1) = frac{omega}{4 P_0} m r^2 omega_{text{max}}^2 )So,( cos(omega t_1) = 1 - frac{omega}{4 P_0} m r^2 omega_{text{max}}^2 )Substituting back into ( E_{text{batteries}} ):( E_{text{batteries}} = frac{P_0}{omega} ( cos(omega t_1) - cos(omega T) ) = frac{P_0}{omega} left( 1 - frac{omega}{4 P_0} m r^2 omega_{text{max}}^2 - cos(omega T) right ) )Simplify:( E_{text{batteries}} = frac{P_0}{omega} - frac{m r^2 omega_{text{max}}^2}{4} - frac{P_0}{omega} cos(omega T) )But ( frac{P_0}{omega} (1 - cos(omega T)) ) is the total energy, so:( E_{text{batteries}} = E_{text{total}} - frac{m r^2 omega_{text{max}}^2}{4} )Wait, let me check:( E_{text{total}} = frac{P_0}{omega} (1 - cos(omega T)) )So,( E_{text{batteries}} = E_{text{total}} - E_{text{flywheel}} = frac{P_0}{omega} (1 - cos(omega T)) - frac{1}{4} m r^2 omega_{text{max}}^2 )But from earlier, ( E_{text{flywheel}} = frac{1}{4} m r^2 omega_{text{max}}^2 = frac{P_0}{omega} (1 - cos(omega t_1)) )So, substituting:( E_{text{batteries}} = frac{P_0}{omega} (1 - cos(omega T)) - frac{P_0}{omega} (1 - cos(omega t_1)) = frac{P_0}{omega} ( cos(omega t_1) - cos(omega T) ) )Which is consistent.Alternatively, using the expression for ( cos(omega t_1) ):( E_{text{batteries}} = frac{P_0}{omega} left( 1 - frac{omega}{4 P_0} m r^2 omega_{text{max}}^2 - cos(omega T) right ) )But perhaps it's more straightforward to leave it as:( C = frac{P_0}{omega} ( cos(omega t_1) - cos(omega T) ) )But since ( t_1 ) is expressed in terms of ( omega_{text{max}} ), we can substitute ( cos(omega t_1) ) as ( 1 - frac{omega}{4 P_0} m r^2 omega_{text{max}}^2 ), so:( C = frac{P_0}{omega} left( 1 - frac{omega}{4 P_0} m r^2 omega_{text{max}}^2 - cos(omega T) right ) )Simplify:( C = frac{P_0}{omega} - frac{m r^2 omega_{text{max}}^2}{4} - frac{P_0}{omega} cos(omega T) )Alternatively, factor out ( frac{P_0}{omega} ):( C = frac{P_0}{omega} left( 1 - cos(omega T) - frac{omega}{4 P_0} m r^2 omega_{text{max}}^2 right ) )But I think the most compact form is:( C = frac{P_0}{omega} ( cos(omega t_1) - cos(omega T) ) )Where ( t_1 = frac{2}{omega} arcsinleft( frac{omega_{text{max}} r sqrt{m omega}}{2 sqrt{2 P_0}} right) )Alternatively, if we want to express ( C ) without ( t_1 ), we can use the expression for ( cos(omega t_1) ):( C = frac{P_0}{omega} left( 1 - frac{omega}{4 P_0} m r^2 omega_{text{max}}^2 - cos(omega T) right ) )But let me check the units:( P_0 ) has units of power (W), ( omega ) is angular frequency (rad/s), so ( frac{P_0}{omega} ) has units of energy (J). Similarly, ( m r^2 omega_{text{max}}^2 ) has units of kg m² (rad²/s²) = kg m²/s² = J. So, all terms are in Joules, which is correct.Alternatively, perhaps express ( C ) as:( C = frac{P_0}{omega} (1 - cos(omega T)) - frac{1}{4} m r^2 omega_{text{max}}^2 )Which is ( E_{text{total}} - E_{text{flywheel}} ), which makes sense.So, in conclusion, the required storage capacity ( C ) is the total energy generated minus the energy stored in the flywheel up to ( t_1 ), which is:( C = frac{P_0}{omega} (1 - cos(omega T)) - frac{1}{4} m r^2 omega_{text{max}}^2 )Alternatively, written as:( C = frac{P_0}{omega} ( cos(omega t_1) - cos(omega T) ) )But since ( t_1 ) is a function of ( omega_{text{max}} ), it's probably better to express ( C ) in terms of ( omega_{text{max}} ) as above.So, summarizing:1. The angular velocity of the flywheel as a function of time is:( omega_f(t) = frac{2 sqrt{2 P_0}}{r sqrt{m omega}} sinleft( frac{omega t}{2} right) )2. The required storage capacity of the batteries is:( C = frac{P_0}{omega} (1 - cos(omega T)) - frac{1}{4} m r^2 omega_{text{max}}^2 )Alternatively, using the expression for ( cos(omega t_1) ):( C = frac{P_0}{omega} left( 1 - frac{omega}{4 P_0} m r^2 omega_{text{max}}^2 - cos(omega T) right ) )I think either form is acceptable, but perhaps the first form is more straightforward.</think>

🧠:<think>Alright, so I have two optimization problems to solve here, both involving the same constraint ( x + y + z = 1 ). The first one is about finding the critical points of a function ( f(x, y, z) ) using Lagrange multipliers, and the second is minimizing a cost function ( g(x, y, z) ) under the same constraint. Let me tackle them one by one.Starting with the first problem: finding the critical points of ( f(x, y, z) = ax^2 + by^2 + cz^2 + dxy + exz + fyz ) with the constraint ( x + y + z = 1 ). Since it's a constrained optimization problem, the method of Lagrange multipliers is the way to go.I remember that the method involves introducing a multiplier (λ) for the constraint and then taking the partial derivatives of the function with respect to each variable and setting them equal to zero. So, I need to set up the Lagrangian function:( mathcal{L}(x, y, z, lambda) = ax^2 + by^2 + cz^2 + dxy + exz + fyz - lambda(x + y + z - 1) )Now, I need to find the partial derivatives of ( mathcal{L} ) with respect to x, y, z, and λ, and set each equal to zero.Let's compute each partial derivative:1. Partial derivative with respect to x:( frac{partial mathcal{L}}{partial x} = 2ax + dy + ez - lambda = 0 )2. Partial derivative with respect to y:( frac{partial mathcal{L}}{partial y} = 2by + dx + fz - lambda = 0 )3. Partial derivative with respect to z:( frac{partial mathcal{L}}{partial z} = 2cz + ex + fy - lambda = 0 )4. Partial derivative with respect to λ:( frac{partial mathcal{L}}{partial lambda} = -(x + y + z - 1) = 0 )Which simplifies to:( x + y + z = 1 )So now I have four equations:1. ( 2ax + dy + ez = lambda ) -- (1)2. ( dx + 2by + fz = lambda ) -- (2)3. ( ex + fy + 2cz = lambda ) -- (3)4. ( x + y + z = 1 ) -- (4)My goal is to solve this system of equations for x, y, z, and λ.Let me write equations (1), (2), and (3) as:1. ( 2a x + d y + e z = lambda )2. ( d x + 2b y + f z = lambda )3. ( e x + f y + 2c z = lambda )Since all three equal λ, I can set them equal to each other:From (1) and (2):( 2a x + d y + e z = d x + 2b y + f z )Let me rearrange terms:( (2a - d)x + (d - 2b)y + (e - f)z = 0 ) -- (5)Similarly, from (2) and (3):( d x + 2b y + f z = e x + f y + 2c z )Rearranging:( (d - e)x + (2b - f)y + (f - 2c)z = 0 ) -- (6)And from (1) and (3):( 2a x + d y + e z = e x + f y + 2c z )Rearranging:( (2a - e)x + (d - f)y + (e - 2c)z = 0 ) -- (7)So now I have three equations (5), (6), (7) and the constraint (4). This seems a bit complicated, but maybe I can express this as a linear system.Let me write equations (5), (6), (7) in matrix form:Equation (5): ( (2a - d)x + (d - 2b)y + (e - f)z = 0 )Equation (6): ( (d - e)x + (2b - f)y + (f - 2c)z = 0 )Equation (7): ( (2a - e)x + (d - f)y + (e - 2c)z = 0 )So, the system is:[begin{bmatrix}2a - d & d - 2b & e - f d - e & 2b - f & f - 2c 2a - e & d - f & e - 2c end{bmatrix}begin{bmatrix}x y z end{bmatrix}=begin{bmatrix}0 0 0 end{bmatrix}]This is a homogeneous system. For a non-trivial solution (since x, y, z are not all zero, as they sum to 1), the determinant of the coefficient matrix must be zero. However, without knowing the specific values of a, b, c, d, e, f, it's hard to compute the determinant. Maybe instead, I can express variables in terms of each other.Alternatively, perhaps I can subtract equations to eliminate variables.Looking at equations (5) and (6):Equation (5): ( (2a - d)x + (d - 2b)y + (e - f)z = 0 )Equation (6): ( (d - e)x + (2b - f)y + (f - 2c)z = 0 )Let me subtract equation (6) from equation (5):( [2a - d - (d - e)]x + [d - 2b - (2b - f)]y + [e - f - (f - 2c)]z = 0 )Simplify each term:For x: ( 2a - d - d + e = 2a - 2d + e )For y: ( d - 2b - 2b + f = d - 4b + f )For z: ( e - f - f + 2c = e - 2f + 2c )So, the equation becomes:( (2a - 2d + e)x + (d - 4b + f)y + (e - 2f + 2c)z = 0 ) -- (8)Similarly, subtract equation (7) from equation (5):Equation (5): ( (2a - d)x + (d - 2b)y + (e - f)z = 0 )Equation (7): ( (2a - e)x + (d - f)y + (e - 2c)z = 0 )Subtracting (7) from (5):( [2a - d - (2a - e)]x + [d - 2b - (d - f)]y + [e - f - (e - 2c)]z = 0 )Simplify each term:For x: ( 2a - d - 2a + e = -d + e )For y: ( d - 2b - d + f = -2b + f )For z: ( e - f - e + 2c = -f + 2c )So, the equation becomes:( (-d + e)x + (-2b + f)y + (-f + 2c)z = 0 ) -- (9)Now, equation (9) is:( (e - d)x + (f - 2b)y + (2c - f)z = 0 )Looking at equation (6):( (d - e)x + (2b - f)y + (f - 2c)z = 0 )Notice that equation (9) is just equation (6) multiplied by -1:Equation (6): ( (d - e)x + (2b - f)y + (f - 2c)z = 0 )Multiply by -1: ( (e - d)x + (f - 2b)y + (2c - f)z = 0 ), which is exactly equation (9). So, equations (6) and (9) are dependent, meaning they are the same equation.Similarly, if I subtract other pairs, I might find dependencies.So, essentially, we have two independent equations from (5), (6), (7), and the constraint (4). Let me see.Alternatively, perhaps it's better to express variables in terms of each other.Let me consider equations (1), (2), (3):Equation (1): ( 2a x + d y + e z = lambda )Equation (2): ( d x + 2b y + f z = lambda )Equation (3): ( e x + f y + 2c z = lambda )Since all equal to λ, I can set them equal to each other:From (1) = (2):( 2a x + d y + e z = d x + 2b y + f z )Which simplifies to:( (2a - d)x + (d - 2b)y + (e - f)z = 0 ) -- same as equation (5)Similarly, from (2) = (3):( d x + 2b y + f z = e x + f y + 2c z )Simplify:( (d - e)x + (2b - f)y + (f - 2c)z = 0 ) -- same as equation (6)So, equations (5) and (6) are the two independent equations, and equation (7) is dependent.Therefore, we have two equations (5) and (6) plus the constraint (4). So, three equations for three variables x, y, z.Let me write them again:1. ( (2a - d)x + (d - 2b)y + (e - f)z = 0 ) -- (5)2. ( (d - e)x + (2b - f)y + (f - 2c)z = 0 ) -- (6)3. ( x + y + z = 1 ) -- (4)So, now I need to solve this system.Let me denote the coefficients for clarity:Equation (5):( A x + B y + C z = 0 )Where:A = 2a - dB = d - 2bC = e - fEquation (6):( D x + E y + F z = 0 )Where:D = d - eE = 2b - fF = f - 2cEquation (4):( x + y + z = 1 )So, now we can write:From equation (5): A x + B y + C z = 0From equation (6): D x + E y + F z = 0From equation (4): x + y + z = 1We can solve this system using substitution or matrix methods.Let me express z from equation (4): z = 1 - x - yThen substitute z into equations (5) and (6):Equation (5):A x + B y + C (1 - x - y) = 0Which is:(A - C) x + (B - C) y + C = 0Similarly, equation (6):D x + E y + F (1 - x - y) = 0Which is:(D - F) x + (E - F) y + F = 0So now, we have two equations in x and y:1. ( (A - C)x + (B - C)y = -C ) -- (10)2. ( (D - F)x + (E - F)y = -F ) -- (11)Let me substitute A, B, C, D, E, F:A = 2a - dC = e - fSo, A - C = 2a - d - (e - f) = 2a - d - e + fSimilarly, B - C = (d - 2b) - (e - f) = d - 2b - e + fSimilarly, D = d - eF = f - 2cSo, D - F = (d - e) - (f - 2c) = d - e - f + 2cE = 2b - fE - F = (2b - f) - (f - 2c) = 2b - f - f + 2c = 2b - 2f + 2cSo, equation (10):( (2a - d - e + f)x + (d - 2b - e + f)y = -(e - f) )Equation (11):( (d - e - f + 2c)x + (2b - 2f + 2c)y = -(f - 2c) )This is getting quite messy, but let's proceed.Let me denote:Equation (10):( M x + N y = P )Where:M = 2a - d - e + fN = d - 2b - e + fP = -(e - f) = f - eEquation (11):( Q x + R y = S )Where:Q = d - e - f + 2cR = 2b - 2f + 2cS = -(f - 2c) = 2c - fSo, now we have:M x + N y = PQ x + R y = SWe can solve this system using Cramer's rule or substitution.Let me write it as:[begin{cases}M x + N y = P Q x + R y = Send{cases}]The determinant of the system is:Δ = M R - N QIf Δ ≠ 0, we can find a unique solution.Compute Δ:Δ = M R - N QSubstitute M, N, Q, R:Δ = (2a - d - e + f)(2b - 2f + 2c) - (d - 2b - e + f)(d - e - f + 2c)This is a bit complicated, but let's try to expand it.First term: (2a - d - e + f)(2b - 2f + 2c)Let me factor out 2 from the second bracket:= (2a - d - e + f)*2*(b - f + c)= 2*(2a - d - e + f)*(b - f + c)Similarly, the second term: (d - 2b - e + f)(d - e - f + 2c)Let me write it as:= (d - 2b - e + f)*(d - e - f + 2c)Hmm, this is getting too involved. Maybe instead of expanding, I can keep it symbolic.Alternatively, perhaps I can express x and y in terms of the determinants.Using Cramer's rule:x = (P R - N S) / Δy = (M S - P Q) / ΔBut this will involve a lot of substitutions.Alternatively, maybe I can express y from equation (10) and substitute into equation (11).From equation (10):M x + N y = P => y = (P - M x)/NSubstitute into equation (11):Q x + R*(P - M x)/N = SMultiply both sides by N:Q N x + R (P - M x) = S NExpand:Q N x + R P - R M x = S NFactor x:x (Q N - R M) + R P = S NThus,x = (S N - R P) / (Q N - R M)Similarly, once x is found, y can be found from equation (10), and then z from equation (4).But given the complexity, perhaps it's better to leave the solution in terms of the coefficients.Alternatively, perhaps I can write the solution as:x = [ (f - e)(2b - 2f + 2c) - (d - 2b - e + f)(2c - f) ] / ΔSimilarly for y.But honestly, without specific values for a, b, c, d, e, f, it's impossible to simplify further. Therefore, the critical points are given by solving the system above, which can be expressed in terms of the coefficients.So, in conclusion, the critical points are the solutions to the system:1. ( (2a - d - e + f)x + (d - 2b - e + f)y = f - e )2. ( (d - e - f + 2c)x + (2b - 2f + 2c)y = 2c - f )3. ( x + y + z = 1 )Which can be solved for x, y, z in terms of a, b, c, d, e, f.Now, moving on to the second problem: minimizing the cost function ( g(x, y, z) = 3x^2 + 4y^2 + 5z^2 - 2xy - 3xz ) with the same constraint ( x + y + z = 1 ).Again, this is a constrained optimization problem, so I'll use Lagrange multipliers.Set up the Lagrangian:( mathcal{L}(x, y, z, lambda) = 3x^2 + 4y^2 + 5z^2 - 2xy - 3xz - lambda(x + y + z - 1) )Compute the partial derivatives:1. Partial derivative with respect to x:( frac{partial mathcal{L}}{partial x} = 6x - 2y - 3z - lambda = 0 ) -- (12)2. Partial derivative with respect to y:( frac{partial mathcal{L}}{partial y} = 8y - 2x - lambda = 0 ) -- (13)3. Partial derivative with respect to z:( frac{partial mathcal{L}}{partial z} = 10z - 3x - lambda = 0 ) -- (14)4. Partial derivative with respect to λ:( frac{partial mathcal{L}}{partial lambda} = -(x + y + z - 1) = 0 )Which simplifies to:( x + y + z = 1 ) -- (15)So, the system of equations is:12. ( 6x - 2y - 3z = lambda )13. ( -2x + 8y = lambda )14. ( -3x + 10z = lambda )15. ( x + y + z = 1 )Now, let's solve this system.From equations (12), (13), (14), all equal to λ, so we can set them equal to each other.First, set equation (12) equal to equation (13):( 6x - 2y - 3z = -2x + 8y )Bring all terms to left:( 6x + 2x - 2y - 8y - 3z = 0 )Simplify:( 8x - 10y - 3z = 0 ) -- (16)Similarly, set equation (13) equal to equation (14):( -2x + 8y = -3x + 10z )Bring all terms to left:( -2x + 3x + 8y - 10z = 0 )Simplify:( x + 8y - 10z = 0 ) -- (17)Now, we have equations (16), (17), and (15):16. ( 8x - 10y - 3z = 0 )17. ( x + 8y - 10z = 0 )15. ( x + y + z = 1 )Let me write these equations:Equation (16): 8x -10y -3z = 0Equation (17): x +8y -10z = 0Equation (15): x + y + z = 1Let me try to solve this system.First, from equation (15): z = 1 - x - ySubstitute z into equations (16) and (17):Equation (16):8x -10y -3(1 - x - y) = 0Expand:8x -10y -3 + 3x + 3y = 0Combine like terms:(8x + 3x) + (-10y + 3y) -3 = 011x -7y -3 = 0So, 11x -7y = 3 -- (18)Equation (17):x +8y -10(1 - x - y) = 0Expand:x +8y -10 +10x +10y = 0Combine like terms:(1x +10x) + (8y +10y) -10 = 011x +18y -10 = 0So, 11x +18y =10 -- (19)Now, we have:Equation (18): 11x -7y = 3Equation (19): 11x +18y =10Subtract equation (18) from equation (19):(11x +18y) - (11x -7y) =10 -3Simplify:11x +18y -11x +7y =725y =7Thus, y = 7/25Now, substitute y =7/25 into equation (18):11x -7*(7/25) =3Compute 7*(7/25)=49/25So,11x -49/25 =3Convert 3 to 75/25:11x =75/25 +49/25 =124/25Thus, x= (124/25)/11 =124/(25*11)=124/275Simplify 124/275: divide numerator and denominator by GCD(124,275). GCD is 1, so x=124/275Now, from equation (15): z=1 -x -y=1 -124/275 -7/25Convert 7/25 to 77/275:z=1 -124/275 -77/275= (275/275 -124/275 -77/275)= (275 -124 -77)/275=(275 -201)/275=74/275Simplify 74/275: divide numerator and denominator by GCD(74,275)=1, so z=74/275Thus, the solution is:x=124/275, y=7/25, z=74/275Let me check if these satisfy equation (17):x +8y -10z=124/275 +8*(7/25) -10*(74/275)Compute 8*(7/25)=56/25=56*11/275=616/27510*(74/275)=740/275So,124/275 +616/275 -740/275=(124 +616 -740)/275=(740 -740)/275=0Yes, it satisfies.Similarly, check equation (16):8x -10y -3z=8*(124/275) -10*(7/25) -3*(74/275)Compute:8*(124/275)=992/27510*(7/25)=70/25=70*11/275=770/2753*(74/275)=222/275So,992/275 -770/275 -222/275=(992 -770 -222)/275=(992 -992)/275=0Good.And equation (15): x+y+z=124/275 +7/25 +74/275=124/275 +77/275 +74/275=(124+77+74)/275=275/275=1Perfect.So, the values are:x=124/275, y=7/25, z=74/275Simplify fractions:124/275: 124 and 275 have GCD 1, so it's 124/2757/25 is already simplified74/275: GCD 1, so 74/275Alternatively, as decimals:124/275 ≈0.45097/25=0.2874/275≈0.2691Check if they sum to 1: 0.4509+0.28+0.2691≈1.0Yes.So, the minimal cost occurs at x≈0.4509, y≈0.28, z≈0.2691But since the question asks for exact values, we'll keep them as fractions.So, x=124/275, y=7/25, z=74/275Alternatively, we can write them with a common denominator:124/275, 77/275, 74/275But 7/25 is 77/275, so that's fine.So, the minimal cost is achieved at x=124/275, y=77/275, z=74/275.Wait, actually, 7/25 is 77/275, so yes, all in 275 denominator.Alternatively, we can write them as:x=124/275, y=7/25, z=74/275But 7/25 is 77/275, so it's consistent.So, the values are x=124/275, y=7/25, z=74/275.I think that's the solution.</think>

🧠:<think>Okay, so I need to help this mother plan her trip to Sesame Place with her two kids. She wants to minimize the total time spent, which includes both waiting in lines for rides and walking between them. There are 10 major rides, each with different wait times and varying distances between them. Plus, her kids have favorite rides—Ride 3 and Ride 7—that they must visit. Also, the total time shouldn't exceed 6 hours. Hmm, that sounds like an optimization problem. Let me break this down step by step.First, I need to understand what exactly needs to be optimized. The goal is to minimize the total time, which is the sum of waiting times and walking times. So, for each ride, there's a wait time ( w_i ), and between any two rides ( i ) and ( j ), there's a walking distance ( d_{ij} ). Since walking speed is 1 meter per minute, the walking time between rides will just be equal to the distance in meters.So, if we think about the problem, it's similar to the Traveling Salesman Problem (TSP), where we need to find the shortest possible route that visits each city (or in this case, ride) exactly once and returns to the origin. However, in this case, we don't necessarily have to visit all rides, but the mother might want to maximize the number of rides within the time constraint. Wait, actually, the problem doesn't specify whether she wants to visit all rides or just a subset. Hmm, the problem says "the park has a total of 10 major rides," but it doesn't say she has to visit all of them. So, maybe she can choose which rides to visit, but she must include Ride 3 and Ride 7 because her kids want to go on them.So, the optimization problem is about selecting a subset of rides that includes Ride 3 and Ride 7, and then determining the order to visit them such that the total time (waiting plus walking) is minimized, and the total time doesn't exceed 6 hours.But wait, the problem says "formulate an optimization problem to minimize the total time spent in the park." So, maybe she wants to visit all 10 rides? Or is it possible that she can choose which ones to visit? Hmm, the wording is a bit ambiguous. Let me check the original problem again.It says, "the park has a total of 10 major rides, each with varying wait times and distances between them." Then, in part 1, she wants to minimize the total time, which includes both walking and waiting. So, perhaps she wants to visit all 10 rides, but in an optimal order. But in part 2, it mentions that the children have favorite rides they must visit: Ride 3 and Ride 7. So, maybe she can choose which rides to visit, but must include Ride 3 and Ride 7, and the total time shouldn't exceed 6 hours.Wait, the problem says "the park has a total of 10 major rides," but it doesn't specify that she has to visit all of them. So, perhaps she can choose a subset that includes Ride 3 and Ride 7, and then find the optimal route for that subset. But then, the total time should be minimized, and the total time shouldn't exceed 6 hours.Alternatively, maybe she wants to visit all 10 rides, but the problem is to find the order that minimizes the total time, including walking and waiting, with the constraints that Ride 3 and Ride 7 must be included, and the total time is under 6 hours.Wait, the problem says "the park has a total of 10 major rides," but it doesn't say she has to visit all of them. So, perhaps she can choose which rides to visit, but must include Ride 3 and Ride 7. Then, the optimization is to select a subset of rides (including 3 and 7) and find the order to visit them such that the total time is minimized, and the total time is under 6 hours.But then, the problem says "formulate an optimization problem to minimize the total time spent in the park." So, maybe she wants to visit as many rides as possible within 6 hours, but that's not explicitly stated. Hmm, this is a bit confusing.Wait, let's re-examine the problem statement:1. The mother decides to minimize the total time spent, which includes both walking and waiting times. If each ride (i) has a wait time (w_i) (in minutes) and the walking distance between ride (i) and ride (j) is represented by (d_{ij}) (in meters), formulate an optimization problem to minimize the total time spent in the park. Assume that the walking speed is constant at 1 meter per minute.2. Given that her two children have favorite rides they must visit: Ride 3 and Ride 7, include these as constraints in your optimization problem. Additionally, ensure that the total time spent in the park does not exceed 6 hours.So, part 1 is about formulating the optimization problem without considering the constraints. Then, part 2 adds constraints: must visit Ride 3 and Ride 7, and total time <= 6 hours.So, in part 1, the problem is to find the order of visiting rides (maybe all 10) that minimizes the total time, which is the sum of waiting times and walking times. Then, in part 2, we add constraints that Ride 3 and Ride 7 must be included, and the total time must not exceed 6 hours.But wait, the problem doesn't specify whether she has to visit all rides or not. So, perhaps in part 1, she can choose any subset of rides, but in part 2, she must include Ride 3 and Ride 7, and the total time must be <= 6 hours.Alternatively, maybe she has to visit all 10 rides, and in part 2, she must include Ride 3 and Ride 7, but since she's already visiting all rides, that's automatically satisfied. Hmm, but the problem says "include these as constraints," which suggests that in part 1, they weren't necessarily included.So, perhaps in part 1, the optimization is over all possible subsets of rides (including all 10), and in part 2, we add the constraints that Ride 3 and Ride 7 must be included, and total time <= 6 hours.But the problem is a bit unclear. Maybe I should assume that she wants to visit all 10 rides, and in part 2, she must include Ride 3 and Ride 7, but since she's already visiting all rides, that's redundant. Hmm, that doesn't make sense.Alternatively, maybe she can choose which rides to visit, but must include Ride 3 and Ride 7, and the total time must be <= 6 hours. So, in part 1, the problem is to minimize the total time, which includes waiting and walking, without any constraints on specific rides. Then, in part 2, we add the constraints that Ride 3 and Ride 7 must be included, and total time <= 6 hours.But the problem says "the park has a total of 10 major rides," so maybe she can choose to visit any number of them, but must include Ride 3 and Ride 7, and the total time must be <= 6 hours.Wait, the problem is a bit ambiguous, but perhaps the key is that in part 1, it's about formulating the optimization problem without constraints, and in part 2, we add the constraints.So, for part 1, the optimization problem is to find a route through some subset of the 10 rides, starting and ending at the entrance (or some starting point), such that the total time (waiting + walking) is minimized.But actually, the problem doesn't specify a starting point. So, perhaps the mother starts at the entrance, visits some rides, and ends at the entrance? Or maybe she can end anywhere? Hmm, that's another ambiguity.Wait, in most amusement parks, you can start and end anywhere, but for the purpose of this problem, maybe we can assume that she starts at a specific point, say the entrance, and then visits the rides in some order, and then returns to the entrance? Or maybe not, since the problem doesn't specify.Alternatively, maybe the walking distance is between rides, so the total walking time is the sum of the distances between consecutive rides in the route. So, if she visits rides in the order ( r_1, r_2, ..., r_n ), then the total walking time is ( d_{r_1 r_2} + d_{r_2 r_3} + ... + d_{r_{n-1} r_n} ). And the total waiting time is the sum of ( w_{r_i} ) for each ride ( r_i ).So, the total time is the sum of all waiting times plus the sum of all walking times between consecutive rides.Therefore, the problem is to choose a subset of rides (including Ride 3 and Ride 7 in part 2) and an order to visit them such that the total time is minimized, and in part 2, the total time is <= 6 hours.But wait, in part 1, she just wants to minimize the total time, which could be interpreted as visiting all rides, but the problem doesn't specify. So, perhaps in part 1, the optimization is over all possible subsets of rides, and the objective is to minimize the total time, which includes waiting and walking. Then, in part 2, we add constraints that Ride 3 and Ride 7 must be included, and the total time must be <= 6 hours.Alternatively, maybe she wants to visit all 10 rides, and the problem is to find the order that minimizes the total time, including walking and waiting. Then, in part 2, we add the constraints that Ride 3 and Ride 7 must be included (which they already are since she's visiting all rides) and the total time must be <= 6 hours.But that seems redundant because if she's visiting all rides, then Ride 3 and Ride 7 are already included. So, perhaps the problem is that she can choose which rides to visit, but must include Ride 3 and Ride 7, and the total time must be <= 6 hours.Given that, I think the problem is that she can choose a subset of rides, including Ride 3 and Ride 7, and find the order to visit them such that the total time (waiting + walking) is minimized, and the total time is <= 6 hours.So, to formulate this, we can model it as a variation of the Traveling Salesman Problem with Time Windows or the Vehicle Routing Problem, but in this case, it's a single route with a time constraint.Let me try to define the variables and constraints.Let me denote:- ( n = 10 ) rides, indexed from 1 to 10.- ( w_i ) = wait time for ride ( i ), in minutes.- ( d_{ij} ) = distance between ride ( i ) and ride ( j ), in meters. Since walking speed is 1 m/min, the walking time is equal to ( d_{ij} ) minutes.We need to decide which rides to visit (including Ride 3 and Ride 7) and the order to visit them such that the total time is minimized and <= 6 hours (360 minutes).So, the decision variables are:- ( x_i ): binary variable indicating whether ride ( i ) is visited (1) or not (0).- ( y_{ij} ): binary variable indicating whether the route goes from ride ( i ) to ride ( j ) (1) or not (0).But since this is a single route, we need to ensure that each ride is entered and exited exactly once, except for the start and end points. However, since the mother can start and end anywhere, it's a bit more complex.Alternatively, we can model this as a permutation problem where we select a subset of rides and find the optimal order to visit them.But this is getting complicated. Maybe a better way is to use integer programming.Let me define:- ( x_i in {0,1} ): 1 if ride ( i ) is visited, 0 otherwise.- ( t_i ): the time at which ride ( i ) is visited.But this might be too complex.Alternatively, we can use the following formulation:Minimize ( sum_{i=1}^{10} w_i x_i + sum_{i=1}^{10} sum_{j=1}^{10} d_{ij} y_{ij} )Subject to:1. ( x_3 = 1 ), ( x_7 = 1 ) (must visit Ride 3 and Ride 7).2. For each ride ( i ), if ( x_i = 1 ), then the number of incoming edges ( sum_{j=1}^{10} y_{ji} = x_i ).3. For each ride ( i ), if ( x_i = 1 ), then the number of outgoing edges ( sum_{j=1}^{10} y_{ij} = x_i ).4. The total time ( sum_{i=1}^{10} w_i x_i + sum_{i=1}^{10} sum_{j=1}^{10} d_{ij} y_{ij} leq 360 ) minutes.5. ( y_{ij} leq x_i ) for all ( i, j ).6. ( y_{ij} leq x_j ) for all ( i, j ).7. ( x_i in {0,1} ), ( y_{ij} in {0,1} ).But this is a bit abstract. Let me try to write it more formally.Let me define:- ( x_i in {0,1} ): 1 if ride ( i ) is visited, 0 otherwise.- ( y_{ij} in {0,1} ): 1 if the route goes from ride ( i ) to ride ( j ), 0 otherwise.The objective function is:Minimize ( sum_{i=1}^{10} w_i x_i + sum_{i=1}^{10} sum_{j=1}^{10} d_{ij} y_{ij} )Subject to:1. ( x_3 = 1 ), ( x_7 = 1 ).2. For each ride ( i ), ( sum_{j=1}^{10} y_{ji} = x_i ) (each visited ride has exactly one incoming edge).3. For each ride ( i ), ( sum_{j=1}^{10} y_{ij} = x_i ) (each visited ride has exactly one outgoing edge).4. The total time ( sum_{i=1}^{10} w_i x_i + sum_{i=1}^{10} sum_{j=1}^{10} d_{ij} y_{ij} leq 360 ).5. ( y_{ij} leq x_i ) for all ( i, j ).6. ( y_{ij} leq x_j ) for all ( i, j ).Additionally, we need to ensure that the route forms a single path without cycles, which is tricky. To prevent cycles, we can use the Miller-Tucker-Zemlin (MTZ) constraints, but that might complicate things further.Alternatively, we can assume that the route starts at a specific point, say ride 1, but the problem doesn't specify a starting point. So, perhaps the mother can start at any ride, visit others, and end anywhere.Wait, but if she starts at a ride, she must have an incoming edge to it, unless it's the start. Hmm, this is getting too complex.Maybe a better approach is to model this as a graph where nodes are rides, and edges have weights equal to the walking time between rides. Then, the problem is to find a path that visits a subset of nodes (including Ride 3 and Ride 7) such that the total time (sum of wait times plus sum of walking times) is minimized and <= 360 minutes.But this is essentially the Traveling Salesman Problem with a time constraint and subset selection.Given the complexity, perhaps the problem expects a formulation rather than solving it computationally.So, to answer part 1, the optimization problem can be formulated as follows:Minimize ( sum_{i=1}^{10} w_i x_i + sum_{i=1}^{10} sum_{j=1}^{10} d_{ij} y_{ij} )Subject to:1. ( x_3 = 1 ), ( x_7 = 1 ) (for part 2).2. For each ride ( i ), ( sum_{j=1}^{10} y_{ji} = x_i ).3. For each ride ( i ), ( sum_{j=1}^{10} y_{ij} = x_i ).4. ( sum_{i=1}^{10} w_i x_i + sum_{i=1}^{10} sum_{j=1}^{10} d_{ij} y_{ij} leq 360 ).5. ( y_{ij} leq x_i ) for all ( i, j ).6. ( y_{ij} leq x_j ) for all ( i, j ).7. ( x_i in {0,1} ), ( y_{ij} in {0,1} ).But in part 1, the constraints 1 and 4 are not present. So, in part 1, the problem is to minimize the total time without constraints on specific rides and without the time limit. Then, in part 2, we add the constraints that Ride 3 and Ride 7 must be visited, and the total time must be <= 360 minutes.So, the formulation for part 1 is:Minimize ( sum_{i=1}^{10} w_i x_i + sum_{i=1}^{10} sum_{j=1}^{10} d_{ij} y_{ij} )Subject to:1. For each ride ( i ), ( sum_{j=1}^{10} y_{ji} = x_i ).2. For each ride ( i ), ( sum_{j=1}^{10} y_{ij} = x_i ).3. ( y_{ij} leq x_i ) for all ( i, j ).4. ( y_{ij} leq x_j ) for all ( i, j ).5. ( x_i in {0,1} ), ( y_{ij} in {0,1} ).Then, in part 2, we add:6. ( x_3 = 1 ), ( x_7 = 1 ).7. ( sum_{i=1}^{10} w_i x_i + sum_{i=1}^{10} sum_{j=1}^{10} d_{ij} y_{ij} leq 360 ).This seems like a reasonable formulation.But wait, in the formulation, we have to ensure that the route is a single path without cycles. The constraints 1 and 2 ensure that each visited ride has exactly one incoming and one outgoing edge, which is suitable for a path (not a cycle). However, if the route is a cycle, it would require that the number of incoming and outgoing edges for each node is equal, but since we're allowing the route to start and end anywhere, it's a path, not a cycle.But in reality, the mother can start at any ride, visit others, and end at any ride. So, the route is a path, not necessarily a cycle.Therefore, the constraints 1 and 2 are correct for a path.However, in the case where only one ride is visited (which is Ride 3 and Ride 7 in part 2), the constraints would still hold because ( x_i =1 ) for those rides, and the incoming and outgoing edges would be zero, which is consistent with ( x_i =1 ).Wait, no. If only Ride 3 is visited, then ( x_3 =1 ), and the incoming and outgoing edges would need to be 1, but since there are no other rides, this would be impossible. So, perhaps we need to adjust the constraints.Alternatively, maybe the route must start and end at specific points, but the problem doesn't specify. Hmm, this is getting too detailed.Alternatively, perhaps the problem is intended to be a TSP where all rides are visited, and the mother starts and ends at the same point. But the problem doesn't specify that she has to return to the starting point.Given the ambiguity, perhaps the problem expects a formulation where the mother visits all rides, starting and ending at the entrance, but that's not specified.Alternatively, maybe the problem is intended to be a simple TSP without considering subsets, but the problem mentions that the mother can choose which rides to visit, given the constraints.Given that, perhaps the formulation I provided earlier is acceptable, even though it's a bit complex.So, to summarize, the optimization problem is an integer linear program where we decide which rides to visit (including Ride 3 and Ride 7 in part 2), and the order to visit them, such that the total time (waiting + walking) is minimized and does not exceed 6 hours.Therefore, the formulation is as follows:Minimize ( sum_{i=1}^{10} w_i x_i + sum_{i=1}^{10} sum_{j=1}^{10} d_{ij} y_{ij} )Subject to:1. ( x_3 = 1 ), ( x_7 = 1 ) (for part 2).2. For each ride ( i ), ( sum_{j=1}^{10} y_{ji} = x_i ).3. For each ride ( i ), ( sum_{j=1}^{10} y_{ij} = x_i ).4. ( sum_{i=1}^{10} w_i x_i + sum_{i=1}^{10} sum_{j=1}^{10} d_{ij} y_{ij} leq 360 ) (for part 2).5. ( y_{ij} leq x_i ) for all ( i, j ).6. ( y_{ij} leq x_j ) for all ( i, j ).7. ( x_i in {0,1} ), ( y_{ij} in {0,1} ).This should be the formulation for the optimization problem.Now, to optimize the plan, we would need to solve this integer linear program. However, since this is a theoretical problem, we can't compute the exact solution here, but we can outline the steps:1. Define the decision variables ( x_i ) and ( y_{ij} ).2. Set up the objective function to minimize the total time.3. Add the constraints for flow conservation (each ride has one incoming and one outgoing edge if visited).4. Add the constraints for the specific rides (Ride 3 and Ride 7) and the total time limit.5. Solve the ILP using a solver like CPLEX, Gurobi, or even Excel Solver.6. The solution will give the optimal subset of rides and the order to visit them, minimizing the total time while satisfying the constraints.So, in conclusion, the optimization problem is formulated as an integer linear program with the objective of minimizing the total time, including waiting and walking, with constraints on visiting specific rides and a time limit.</think>

🧠:<think>Okay, so I have this problem about a UAV wing and calculating the total lift. Hmm, the lift distribution is given by this elliptical equation: ( L(y) = L_0 sqrt{1 - left(frac{2y}{b}right)^2} ). I need to find the total lift by integrating this across the entire wingspan. Alright, so the wingspan is from ( y = -b/2 ) to ( y = b/2 ), right? Because the centerline is at y=0, and the wingspan is b, so each side is b/2. So, the integral should be from -b/2 to b/2 of ( L(y) ) dy. Let me write that down: ( L = int_{-b/2}^{b/2} L_0 sqrt{1 - left(frac{2y}{b}right)^2} dy ). Hmm, this looks like an integral that might relate to the area of an ellipse or something. Wait, the integrand is similar to the equation of an ellipse. Let me think. If I make a substitution to simplify the integral, maybe a trigonometric substitution or something else.Let me set ( u = frac{2y}{b} ). Then, ( du = frac{2}{b} dy ), so ( dy = frac{b}{2} du ). When y = -b/2, u = -1, and when y = b/2, u = 1. So, substituting, the integral becomes:( L = L_0 int_{-1}^{1} sqrt{1 - u^2} cdot frac{b}{2} du ).That simplifies to ( L = frac{L_0 b}{2} int_{-1}^{1} sqrt{1 - u^2} du ).Now, the integral of ( sqrt{1 - u^2} ) from -1 to 1 is a standard integral. I remember that ( int_{-a}^{a} sqrt{a^2 - u^2} du ) is the area of a semicircle, which is ( frac{pi a^2}{2} ). But here, a is 1, so the integral is ( frac{pi (1)^2}{2} = frac{pi}{2} ).Wait, actually, no. The integral from -1 to 1 of ( sqrt{1 - u^2} du ) is the area of a semicircle with radius 1, which is ( frac{pi}{2} ). But wait, actually, the area of a full circle is ( pi r^2 ), so a semicircle is ( frac{pi r^2}{2} ). But in this case, the integral is from -1 to 1, which is the area of a semicircle, so it's ( frac{pi}{2} ).Wait, no, actually, the integral of ( sqrt{1 - u^2} ) from -1 to 1 is the area of a semicircle, which is ( frac{pi}{2} ). So, plugging that in, we get:( L = frac{L_0 b}{2} cdot frac{pi}{2} = frac{L_0 b pi}{4} ).Wait, but I thought the total lift for an elliptical wing is ( frac{2}{pi} L_0 b ). Hmm, maybe I made a mistake here. Let me check my substitution again.Wait, when I set ( u = frac{2y}{b} ), then when y goes from -b/2 to b/2, u goes from -1 to 1. The integral becomes ( int_{-1}^{1} sqrt{1 - u^2} cdot frac{b}{2} du ). So, that's correct.But the integral of ( sqrt{1 - u^2} ) from -1 to 1 is indeed ( frac{pi}{2} ), because it's the area of a semicircle. So, multiplying by ( frac{b}{2} ), we get ( frac{pi b}{4} ). Then, multiplying by ( L_0 ), we get ( L = frac{pi L_0 b}{4} ).Wait, but I thought the total lift for an elliptical wing is ( frac{2}{pi} L_0 b ). Hmm, maybe I confused something. Let me think again.Alternatively, maybe I should consider that the lift distribution is symmetric, so I can integrate from 0 to b/2 and then double it. Let's try that.So, ( L = 2 int_{0}^{b/2} L_0 sqrt{1 - left(frac{2y}{b}right)^2} dy ).Let me make the substitution ( u = frac{2y}{b} ), so ( du = frac{2}{b} dy ), ( dy = frac{b}{2} du ). When y=0, u=0; when y=b/2, u=1.So, the integral becomes ( 2 cdot L_0 int_{0}^{1} sqrt{1 - u^2} cdot frac{b}{2} du ).Simplify: ( 2 cdot L_0 cdot frac{b}{2} int_{0}^{1} sqrt{1 - u^2} du ).The integral from 0 to 1 of ( sqrt{1 - u^2} du ) is ( frac{pi}{4} ), because it's a quarter of the circle. So, ( frac{pi}{4} ).Thus, ( L = L_0 b cdot frac{pi}{4} ).Wait, that's the same result as before. So, maybe my initial thought about ( frac{2}{pi} L_0 b ) is incorrect. Let me check the formula for lift on an elliptical wing.I recall that for an elliptical lift distribution, the total lift is ( frac{2}{pi} L_0 b ). Hmm, but according to my integration, it's ( frac{pi}{4} L_0 b ). There's a discrepancy here.Wait, perhaps I made a mistake in the substitution. Let me double-check.The original integral is ( int_{-b/2}^{b/2} L_0 sqrt{1 - left(frac{2y}{b}right)^2} dy ).Let me consider the substitution ( u = frac{2y}{b} ), so ( y = frac{b u}{2} ), ( dy = frac{b}{2} du ).So, when y = -b/2, u = -1; y = b/2, u = 1.Thus, the integral becomes ( L_0 cdot frac{b}{2} int_{-1}^{1} sqrt{1 - u^2} du ).As before, the integral is ( frac{pi}{2} ), so total lift is ( L_0 cdot frac{b}{2} cdot frac{pi}{2} = frac{pi L_0 b}{4} ).Hmm, so according to this, the total lift is ( frac{pi}{4} L_0 b ). But I thought it was ( frac{2}{pi} L_0 b ). Maybe I confused the formula.Wait, perhaps the formula for the total lift on an elliptical wing is indeed ( frac{pi}{4} L_0 b ). Let me think about the units. Lift has units of force, which is mass times acceleration, so kg·m/s². The lift distribution is given per unit span, so integrating over span gives total lift.Alternatively, maybe I should consider that the peak lift ( L_0 ) is the lift per unit span at the centerline. So, the total lift would be the integral over the span, which is ( int_{-b/2}^{b/2} L(y) dy ).Wait, but in aerodynamics, the total lift is often given by the integral of the lift distribution over the wingspan. So, if ( L(y) ) is the lift per unit span, then total lift is ( int L(y) dy ).But in this case, is ( L(y) ) the total lift at each y, or is it the lift per unit span? The problem says "lift distribution across the wingspan", so I think it's lift per unit span. So, integrating over the span gives the total lift.So, according to my calculation, it's ( frac{pi}{4} L_0 b ).Wait, but I'm pretty sure that for an elliptical lift distribution, the total lift is ( frac{2}{pi} L_0 b ). Maybe I'm confusing the formula. Let me check.Wait, perhaps the formula is ( L = frac{2}{pi} L_0 b ). So, where did I go wrong?Wait, let me think about the integral again. The integral of ( sqrt{1 - u^2} ) from -1 to 1 is ( frac{pi}{2} ). So, ( L = L_0 cdot frac{b}{2} cdot frac{pi}{2} = frac{pi L_0 b}{4} ).But if the standard formula is ( frac{2}{pi} L_0 b ), then perhaps I made a mistake in the substitution.Wait, maybe the substitution is incorrect. Let me try a different substitution.Let me set ( u = frac{y}{b/2} ), so ( u = frac{2y}{b} ). Then, ( du = frac{2}{b} dy ), so ( dy = frac{b}{2} du ). When y = -b/2, u = -1; y = b/2, u = 1.So, the integral becomes ( L_0 int_{-1}^{1} sqrt{1 - u^2} cdot frac{b}{2} du ).Which is ( frac{L_0 b}{2} cdot frac{pi}{2} = frac{pi L_0 b}{4} ).Hmm, same result. So, maybe the standard formula is different. Alternatively, perhaps the problem defines ( L(y) ) differently.Wait, the problem says "the lift distribution across the wingspan, ( L(y) ), can be approximated by the elliptical equation...". So, maybe in this case, the total lift is indeed ( frac{pi}{4} L_0 b ).Alternatively, perhaps I should consider that the lift distribution is given as ( L(y) = L_0 sqrt{1 - (2y/b)^2} ), and the total lift is the integral over the span.Wait, let me think about the units. If ( L(y) ) is lift per unit span, then integrating over span gives total lift. So, the units would be correct.Alternatively, maybe the problem is using a different scaling. Let me think about the standard elliptical lift distribution.In aerodynamics, the elliptical lift distribution is often given by ( frac{L(y)}{L_0} = sqrt{1 - (y/b)^2} ), where y ranges from -b to b. Wait, but in this problem, y ranges from -b/2 to b/2, so maybe the formula is adjusted accordingly.Wait, in the standard case, the wingspan is 2b, so y ranges from -b to b, and the lift distribution is ( L(y) = L_0 sqrt{1 - (y/b)^2} ). Then, the total lift would be ( int_{-b}^{b} L_0 sqrt{1 - (y/b)^2} dy ).Let me compute that integral. Let ( u = y/b ), so ( du = dy/b ), ( dy = b du ). The integral becomes ( L_0 b int_{-1}^{1} sqrt{1 - u^2} du = L_0 b cdot frac{pi}{2} ).So, total lift is ( frac{pi}{2} L_0 b ).But in our problem, the wingspan is b, and y ranges from -b/2 to b/2. So, the formula is ( L(y) = L_0 sqrt{1 - (2y/b)^2} ). So, the substitution would be similar, but with a factor of 2.So, let me compute the integral again.( L = int_{-b/2}^{b/2} L_0 sqrt{1 - (2y/b)^2} dy ).Let ( u = 2y/b ), so ( du = 2/b dy ), ( dy = (b/2) du ). When y = -b/2, u = -1; y = b/2, u = 1.Thus, ( L = L_0 cdot (b/2) int_{-1}^{1} sqrt{1 - u^2} du = L_0 cdot (b/2) cdot frac{pi}{2} = frac{pi L_0 b}{4} ).So, that's consistent with my earlier result. So, perhaps the standard formula is when the wingspan is 2b, but in this case, the wingspan is b, so the total lift is ( frac{pi}{4} L_0 b ).Alternatively, maybe the problem defines ( L_0 ) differently. Wait, in the standard case, ( L_0 ) is the peak lift per unit span. So, in our problem, ( L_0 ) is the peak lift per unit span at the centerline.Therefore, the total lift is ( frac{pi}{4} L_0 b ).Wait, but I'm still a bit confused because I thought the standard result was ( frac{2}{pi} L_0 b ). Maybe I'm mixing up the definitions.Alternatively, perhaps the problem is using a different scaling factor. Let me think about the integral again.The integral of ( sqrt{1 - u^2} ) from -1 to 1 is ( pi/2 ). So, in our case, the integral is ( frac{pi}{2} ), multiplied by ( L_0 cdot frac{b}{2} ), giving ( frac{pi L_0 b}{4} ).So, unless there's a mistake in my substitution, I think that's the correct result.Wait, let me check the substitution again. If ( u = 2y/b ), then ( y = (b/2)u ), ( dy = (b/2) du ). So, the integral becomes ( L_0 cdot (b/2) int_{-1}^{1} sqrt{1 - u^2} du ).Which is ( L_0 cdot (b/2) cdot frac{pi}{2} = frac{pi L_0 b}{4} ).Yes, that seems correct.So, maybe the answer is ( frac{pi}{4} L_0 b ).Alternatively, perhaps the problem expects the answer in terms of ( frac{2}{pi} L_0 b ), but I think my calculation is correct based on the given formula.So, I'll go with ( L = frac{pi}{4} L_0 b ).Wait, but let me think about the units again. If ( L_0 ) is in N/m (lift per unit span), and b is in meters, then the total lift would be in Newtons, which makes sense.So, I think my answer is correct.Now, moving on to the second problem. The UAV's path is given by ( mathbf{r}(t) = (t^2, sin t, cos t) ). I need to find the curvature ( kappa(t) ) at ( t = pi/4 ).I remember that the curvature of a parametric curve ( mathbf{r}(t) ) is given by:( kappa(t) = frac{||mathbf{r}'(t) times mathbf{r}''(t)||}{||mathbf{r}'(t)||^3} ).So, I need to compute the first and second derivatives of ( mathbf{r}(t) ), then find their cross product, its magnitude, and divide by the cube of the magnitude of the first derivative.Let me compute the first derivative ( mathbf{r}'(t) ):( mathbf{r}'(t) = (2t, cos t, -sin t) ).Then, the second derivative ( mathbf{r}''(t) ):( mathbf{r}''(t) = (2, -sin t, -cos t) ).Now, compute the cross product ( mathbf{r}'(t) times mathbf{r}''(t) ).Using the determinant formula:( mathbf{r}' times mathbf{r}'' = begin{vmatrix} mathbf{i} & mathbf{j} & mathbf{k} 2t & cos t & -sin t 2 & -sin t & -cos t end{vmatrix} ).Calculating the determinant:i component: ( cos t cdot (-cos t) - (-sin t) cdot (-sin t) = -cos^2 t - sin^2 t = -(cos^2 t + sin^2 t) = -1 ).j component: - [2t cdot (-cos t) - (-sin t) cdot 2 ] = - [ -2t cos t + 2 sin t ] = 2t cos t - 2 sin t.k component: 2t cdot (-sin t) - cos t cdot 2 = -2t sin t - 2 cos t.So, the cross product is:( mathbf{r}' times mathbf{r}'' = (-1, 2t cos t - 2 sin t, -2t sin t - 2 cos t) ).Now, compute the magnitude of this cross product:||( mathbf{r}' times mathbf{r}'' )|| = sqrt[ (-1)^2 + (2t cos t - 2 sin t)^2 + (-2t sin t - 2 cos t)^2 ].Let me compute each term:First term: (-1)^2 = 1.Second term: (2t cos t - 2 sin t)^2 = [2(t cos t - sin t)]^2 = 4(t cos t - sin t)^2.Third term: (-2t sin t - 2 cos t)^2 = [ -2(t sin t + cos t) ]^2 = 4(t sin t + cos t)^2.So, the magnitude squared is:1 + 4(t cos t - sin t)^2 + 4(t sin t + cos t)^2.Let me expand the terms inside:First, expand (t cos t - sin t)^2:= t² cos² t - 2t cos t sin t + sin² t.Similarly, (t sin t + cos t)^2:= t² sin² t + 2t sin t cos t + cos² t.So, adding these two:(t² cos² t - 2t cos t sin t + sin² t) + (t² sin² t + 2t sin t cos t + cos² t)= t² (cos² t + sin² t) + (-2t cos t sin t + 2t cos t sin t) + (sin² t + cos² t)= t² (1) + 0 + 1= t² + 1.So, the magnitude squared is:1 + 4(t² + 1) = 1 + 4t² + 4 = 4t² + 5.Therefore, the magnitude is sqrt(4t² + 5).Now, compute ||( mathbf{r}'(t) )||:( mathbf{r}'(t) = (2t, cos t, -sin t) ).So, ||( mathbf{r}'(t) )|| = sqrt[ (2t)^2 + (cos t)^2 + (-sin t)^2 ] = sqrt[4t² + cos² t + sin² t] = sqrt[4t² + 1].So, ||( mathbf{r}'(t) )||^3 = (sqrt(4t² + 1))^3 = (4t² + 1)^(3/2).Therefore, the curvature ( kappa(t) ) is:( kappa(t) = frac{sqrt{4t² + 5}}{(4t² + 1)^{3/2}} ).Simplify this expression:We can write it as ( sqrt{frac{4t² + 5}{(4t² + 1)^3}} ).Alternatively, factor out the denominator:( kappa(t) = frac{sqrt{4t² + 5}}{(4t² + 1)^{3/2}} = frac{sqrt{4t² + 5}}{(4t² + 1) sqrt{4t² + 1}}} = frac{sqrt{4t² + 5}}{(4t² + 1)^{3/2}} ).But perhaps it's better to leave it as is.Now, evaluate this at ( t = pi/4 ).First, compute ( t = pi/4 approx 0.7854 ).Compute 4t²:4*(π/4)^2 = 4*(π²/16) = π²/4 ≈ (9.8696)/4 ≈ 2.4674.So, 4t² + 5 ≈ 2.4674 + 5 = 7.4674.4t² + 1 ≈ 2.4674 + 1 = 3.4674.So, sqrt(4t² + 5) ≈ sqrt(7.4674) ≈ 2.732.(4t² + 1)^(3/2) ≈ (3.4674)^(3/2). Let's compute that.First, sqrt(3.4674) ≈ 1.861.Then, 3.4674 * 1.861 ≈ 3.4674 * 1.861 ≈ let's compute:3 * 1.861 = 5.5830.4674 * 1.861 ≈ approx 0.4674*1.8 ≈ 0.84132, plus 0.4674*0.061 ≈ 0.0285, total ≈ 0.8698.So, total ≈ 5.583 + 0.8698 ≈ 6.4528.So, (4t² + 1)^(3/2) ≈ 6.4528.Thus, curvature κ ≈ 2.732 / 6.4528 ≈ 0.423.But let's compute it more accurately.Alternatively, let's compute it symbolically.We have:( kappa(pi/4) = frac{sqrt{4(pi/4)^2 + 5}}{(4(pi/4)^2 + 1)^{3/2}} ).Simplify:4*(π/4)^2 = (π²)/4.So, numerator inside sqrt: (π²)/4 + 5 = (π² + 20)/4.Denominator inside the power: (π²)/4 + 1 = (π² + 4)/4.So, numerator: sqrt( (π² + 20)/4 ) = (sqrt(π² + 20))/2.Denominator: [ (π² + 4)/4 ]^(3/2) = [ (π² + 4)^(3/2) ] / (4^(3/2)) ) = (π² + 4)^(3/2) / 8.So, overall:( kappa(pi/4) = frac{ (sqrt(π² + 20))/2 }{ (π² + 4)^(3/2) / 8 } = frac{sqrt(π² + 20) * 8}{2 * (π² + 4)^(3/2)} = frac{4 sqrt(π² + 20)}{(π² + 4)^(3/2)} ).We can write this as:( kappa(pi/4) = 4 frac{sqrt{pi² + 20}}{(pi² + 4)^{3/2}} ).Alternatively, factor out the denominator:= 4 / (π² + 4) * sqrt( (π² + 20)/(π² + 4) ).But perhaps it's better to leave it in terms of exponents.Alternatively, we can write it as:= 4 (π² + 20)^{1/2} / (π² + 4)^{3/2} = 4 (π² + 20)^{1/2} (π² + 4)^{-3/2}.But maybe it's better to rationalize or simplify further.Alternatively, compute the numerical value.Compute π² ≈ 9.8696.So, π² + 20 ≈ 29.8696.sqrt(29.8696) ≈ 5.466.π² + 4 ≈ 13.8696.sqrt(13.8696) ≈ 3.724.So, (π² + 4)^(3/2) ≈ (13.8696)^(3/2) = sqrt(13.8696)^3 ≈ 3.724^3 ≈ 3.724 * 3.724 = approx 13.869, then *3.724 ≈ 51.65.Wait, let me compute 3.724^3:3.724 * 3.724 = let's compute:3 * 3 = 93 * 0.724 = 2.1720.724 * 3 = 2.1720.724 * 0.724 ≈ 0.524So, adding up:9 + 2.172 + 2.172 + 0.524 ≈ 13.868.Then, 13.868 * 3.724 ≈ let's compute:13 * 3.724 ≈ 48.4120.868 * 3.724 ≈ approx 3.246Total ≈ 48.412 + 3.246 ≈ 51.658.So, (π² + 4)^(3/2) ≈ 51.658.Numerator: 4 * sqrt(π² + 20) ≈ 4 * 5.466 ≈ 21.864.So, curvature κ ≈ 21.864 / 51.658 ≈ 0.423.So, approximately 0.423.But let's compute it more accurately.Compute numerator: 4 * sqrt(π² + 20).π² ≈ 9.8696, so π² + 20 ≈ 29.8696.sqrt(29.8696) ≈ 5.466.4 * 5.466 ≈ 21.864.Denominator: (π² + 4)^(3/2).π² + 4 ≈ 13.8696.sqrt(13.8696) ≈ 3.724.(13.8696)^(3/2) = (sqrt(13.8696))^3 ≈ 3.724^3.Compute 3.724^3:First, 3.724 * 3.724:3 * 3 = 93 * 0.724 = 2.1720.724 * 3 = 2.1720.724 * 0.724 ≈ 0.524So, total ≈ 9 + 2.172 + 2.172 + 0.524 ≈ 13.868.Then, 13.868 * 3.724:13 * 3.724 = 48.4120.868 * 3.724 ≈ 3.246Total ≈ 48.412 + 3.246 ≈ 51.658.So, denominator ≈ 51.658.Thus, κ ≈ 21.864 / 51.658 ≈ 0.423.So, approximately 0.423.But let's compute it more precisely.Alternatively, use exact expressions.Wait, perhaps we can write it as:( kappa(pi/4) = frac{4 sqrt{pi^2 + 20}}{(pi^2 + 4)^{3/2}} ).Alternatively, factor out the denominator:= ( frac{4}{pi^2 + 4} cdot sqrt{frac{pi^2 + 20}{pi^2 + 4}} ).But I think the answer is better left in terms of π, but perhaps the problem expects a numerical value.Alternatively, compute it more accurately.Compute π ≈ 3.14159265.Compute π² ≈ 9.8696044.So, π² + 20 ≈ 29.8696044.sqrt(29.8696044) ≈ 5.466.4 * 5.466 ≈ 21.864.π² + 4 ≈ 13.8696044.sqrt(13.8696044) ≈ 3.724.(13.8696044)^(3/2) = (sqrt(13.8696044))^3 ≈ 3.724^3.Compute 3.724^3:3.724 * 3.724 = let's compute:3.724 * 3 = 11.1723.724 * 0.724 ≈ 2.696So, total ≈ 11.172 + 2.696 ≈ 13.868.Then, 13.868 * 3.724 ≈ let's compute:13 * 3.724 = 48.4120.868 * 3.724 ≈ 3.246Total ≈ 48.412 + 3.246 ≈ 51.658.So, denominator ≈ 51.658.Thus, κ ≈ 21.864 / 51.658 ≈ 0.423.So, approximately 0.423.But let me compute it more accurately using a calculator.Compute numerator: 4 * sqrt(π² + 20).π² ≈ 9.8696044.So, π² + 20 ≈ 29.8696044.sqrt(29.8696044) ≈ 5.466.4 * 5.466 ≈ 21.864.Denominator: (π² + 4)^(3/2).π² + 4 ≈ 13.8696044.sqrt(13.8696044) ≈ 3.724.(13.8696044)^(3/2) = (3.724)^3 ≈ 51.658.So, κ ≈ 21.864 / 51.658 ≈ 0.423.Alternatively, using more precise calculations:Compute 4 * sqrt(π² + 20):sqrt(29.8696044) ≈ 5.466086.4 * 5.466086 ≈ 21.864344.Denominator: (π² + 4)^(3/2).Compute (13.8696044)^(3/2):First, compute sqrt(13.8696044) ≈ 3.724077.Then, cube it: 3.724077^3.Compute 3.724077 * 3.724077:= (3 + 0.724077)^2 = 9 + 2*3*0.724077 + (0.724077)^2 ≈ 9 + 4.344462 + 0.524 ≈ 13.868462.Then, multiply by 3.724077:13.868462 * 3.724077 ≈ let's compute:13 * 3.724077 ≈ 48.413.0.868462 * 3.724077 ≈ approx 3.246.Total ≈ 48.413 + 3.246 ≈ 51.659.So, denominator ≈ 51.659.Thus, κ ≈ 21.864344 / 51.659 ≈ 0.423.So, approximately 0.423.But let me compute it more precisely:21.864344 / 51.659 ≈Divide 21.864344 by 51.659.51.659 goes into 21.864344 approximately 0.423 times.Yes, so κ ≈ 0.423.But let me check using a calculator for more precision.Alternatively, perhaps the problem expects an exact expression in terms of π, but I think the numerical value is acceptable.So, the curvature at t = π/4 is approximately 0.423.But let me see if I can write it in a simplified exact form.We have:( kappa(pi/4) = frac{4 sqrt{pi^2 + 20}}{(pi^2 + 4)^{3/2}} ).Alternatively, factor out the denominator:= ( frac{4}{pi^2 + 4} cdot sqrt{frac{pi^2 + 20}{pi^2 + 4}} ).But I don't think it simplifies further.So, the exact value is ( frac{4 sqrt{pi^2 + 20}}{(pi^2 + 4)^{3/2}} ), and the approximate value is 0.423.So, I think that's the answer.</think>