🧠:<think>Okay, so I have this problem where I need to find the coefficients of a cubic polynomial that models the number of goals SK Dynamo České Budějovice scored in each game. The polynomial is given by P(x) = ax³ + bx² + cx + d. They provided the number of goals for the first four games: 2, 5, 10, and 17 for x=1, 2, 3, and 4 respectively. Then, I need to use this polynomial to calculate the total goals from x=5 to x=9.Alright, let's start with the first part. I need to find a, b, c, and d. Since it's a cubic polynomial, and we have four points, I can set up a system of equations.For x=1, P(1) = a(1)³ + b(1)² + c(1) + d = a + b + c + d = 2.For x=2, P(2) = a(8) + b(4) + c(2) + d = 8a + 4b + 2c + d = 5.For x=3, P(3) = a(27) + b(9) + c(3) + d = 27a + 9b + 3c + d = 10.For x=4, P(4) = a(64) + b(16) + c(4) + d = 64a + 16b + 4c + d = 17.So, I have four equations:1) a + b + c + d = 22) 8a + 4b + 2c + d = 53) 27a + 9b + 3c + d = 104) 64a + 16b + 4c + d = 17Now, I need to solve this system of equations. Let me write them down again:Equation 1: a + b + c + d = 2Equation 2: 8a + 4b + 2c + d = 5Equation 3: 27a + 9b + 3c + d = 10Equation 4: 64a + 16b + 4c + d = 17I think the best way to solve this is by elimination. Let's subtract Equation 1 from Equation 2, Equation 2 from Equation 3, and Equation 3 from Equation 4 to eliminate d each time.First, subtract Equation 1 from Equation 2:(8a - a) + (4b - b) + (2c - c) + (d - d) = 5 - 2Which simplifies to:7a + 3b + c = 3. Let's call this Equation 5.Next, subtract Equation 2 from Equation 3:(27a - 8a) + (9b - 4b) + (3c - 2c) + (d - d) = 10 - 5Which simplifies to:19a + 5b + c = 5. Let's call this Equation 6.Then, subtract Equation 3 from Equation 4:(64a - 27a) + (16b - 9b) + (4c - 3c) + (d - d) = 17 - 10Simplifies to:37a + 7b + c = 7. Let's call this Equation 7.Now, we have three new equations:Equation 5: 7a + 3b + c = 3Equation 6: 19a + 5b + c = 5Equation 7: 37a + 7b + c = 7Now, let's subtract Equation 5 from Equation 6 to eliminate c:(19a - 7a) + (5b - 3b) + (c - c) = 5 - 3Which is:12a + 2b = 2. Let's call this Equation 8.Similarly, subtract Equation 6 from Equation 7:(37a - 19a) + (7b - 5b) + (c - c) = 7 - 5Simplifies to:18a + 2b = 2. Let's call this Equation 9.Now, we have:Equation 8: 12a + 2b = 2Equation 9: 18a + 2b = 2Subtract Equation 8 from Equation 9:(18a - 12a) + (2b - 2b) = 2 - 2Which is:6a = 0So, 6a = 0 => a = 0.Wait, a is zero? That's interesting. So, the cubic term is zero. So, the polynomial is actually quadratic? Hmm.But let's check if that's correct.If a = 0, then plug back into Equation 8:12(0) + 2b = 2 => 2b = 2 => b = 1.So, b = 1.Now, go back to Equation 5: 7a + 3b + c = 3Since a=0 and b=1, we have:0 + 3(1) + c = 3 => 3 + c = 3 => c = 0.So, c = 0.Now, go back to Equation 1: a + b + c + d = 2Plugging in a=0, b=1, c=0:0 + 1 + 0 + d = 2 => d = 1.So, d = 1.Wait, so the polynomial is P(x) = 0x³ + 1x² + 0x + 1 => P(x) = x² + 1.Let me verify this with the given points.For x=1: 1² + 1 = 2. Correct.x=2: 4 + 1 = 5. Correct.x=3: 9 + 1 = 10. Correct.x=4: 16 + 1 = 17. Correct.So, actually, the polynomial is quadratic, but since it's given as a cubic, maybe the cubic term is zero. So, a=0, b=1, c=0, d=1.Alright, so that's the first part done.Now, moving on to the second part: calculate the total number of goals from x=5 to x=9.So, I need to compute P(5) + P(6) + P(7) + P(8) + P(9).Since P(x) = x² + 1, this is straightforward.Compute each term:P(5) = 25 + 1 = 26P(6) = 36 + 1 = 37P(7) = 49 + 1 = 50P(8) = 64 + 1 = 65P(9) = 81 + 1 = 82Now, sum these up:26 + 37 = 6363 + 50 = 113113 + 65 = 178178 + 82 = 260So, the total number of goals from x=5 to x=9 is 260.Wait, let me double-check the calculations:P(5)=25+1=26P(6)=36+1=37P(7)=49+1=50P(8)=64+1=65P(9)=81+1=82Adding them up:26 + 37 = 6363 + 50 = 113113 + 65 = 178178 + 82 = 260Yes, that seems correct.Alternatively, maybe I can compute the sum using the formula for the sum of squares.Sum from x=5 to x=9 of x² + 1 is equal to sum x² from 5 to 9 plus sum 1 from 5 to 9.Sum x² from 1 to n is n(n+1)(2n+1)/6.So, sum x² from 1 to 9 minus sum x² from 1 to 4.Sum x² from 1 to 9: 9*10*19/6 = (9*10*19)/6 = (1710)/6 = 285.Sum x² from 1 to 4: 4*5*9/6 = (180)/6 = 30.So, sum x² from 5 to 9: 285 - 30 = 255.Sum of 1 from 5 to 9: there are 5 terms, so 5*1=5.Total sum: 255 + 5 = 260.Yes, same result. So, that's correct.Therefore, the total number of goals expected from x=5 to x=9 is 260.Final AnswerThe coefficients are ( a = 0 ), ( b = 1 ), ( c = 0 ), and ( d = 1 ). The total number of goals expected over the next five games is boxed{260}.</think>

🧠:<think>Alright, so I have this problem about Dr. Elena studying two plants, Plant A and Plant B, which are believed to have anti-cancer properties. There are two parts to this problem. Let me try to tackle them one by one.Part 1: Modeling Growth RatesFirst, the problem gives me two differential equations for the growth rates of Plant A and Plant B. The equations are:[frac{dP_A}{dt} = r_A P_A left(1 - frac{P_A}{K_A}right) - alpha P_A P_B][frac{dP_B}{dt} = r_B P_B left(1 - frac{P_B}{K_B}right) - beta P_A P_B]I need to find the equilibrium points and assess their stability. Hmm, okay. So, equilibrium points are where the derivatives are zero, right? That means both (frac{dP_A}{dt} = 0) and (frac{dP_B}{dt} = 0).Let me write down the conditions for equilibrium:1. ( r_A P_A left(1 - frac{P_A}{K_A}right) - alpha P_A P_B = 0 )2. ( r_B P_B left(1 - frac{P_B}{K_B}right) - beta P_A P_B = 0 )So, I need to solve these two equations simultaneously.First, let's consider the trivial equilibrium where both populations are zero. That is, ( P_A = 0 ) and ( P_B = 0 ). Plugging into the equations, both sides become zero, so that's definitely an equilibrium point.Next, let's look for non-trivial equilibria where at least one of the populations is non-zero.Starting with the first equation:( r_A P_A left(1 - frac{P_A}{K_A}right) = alpha P_A P_B )Assuming ( P_A neq 0 ), we can divide both sides by ( P_A ):( r_A left(1 - frac{P_A}{K_A}right) = alpha P_B )Similarly, from the second equation:( r_B P_B left(1 - frac{P_B}{K_B}right) = beta P_A P_B )Assuming ( P_B neq 0 ), divide both sides by ( P_B ):( r_B left(1 - frac{P_B}{K_B}right) = beta P_A )So now, we have two equations:1. ( r_A left(1 - frac{P_A}{K_A}right) = alpha P_B ) --- (1)2. ( r_B left(1 - frac{P_B}{K_B}right) = beta P_A ) --- (2)Let me solve equation (1) for ( P_B ):( P_B = frac{r_A}{alpha} left(1 - frac{P_A}{K_A}right) ) --- (1a)Similarly, solve equation (2) for ( P_A ):( P_A = frac{r_B}{beta} left(1 - frac{P_B}{K_B}right) ) --- (2a)Now, substitute equation (1a) into equation (2a):( P_A = frac{r_B}{beta} left(1 - frac{1}{K_B} cdot frac{r_A}{alpha} left(1 - frac{P_A}{K_A}right) right) )Let me simplify this step by step.First, expand the inner term:( 1 - frac{1}{K_B} cdot frac{r_A}{alpha} left(1 - frac{P_A}{K_A}right) )Let me denote ( frac{r_A}{alpha K_B} ) as a constant, say ( c ). So, it becomes:( 1 - c left(1 - frac{P_A}{K_A}right) )Which is:( 1 - c + frac{c P_A}{K_A} )So, plugging back into equation (2a):( P_A = frac{r_B}{beta} left(1 - c + frac{c P_A}{K_A} right) )Substituting back ( c = frac{r_A}{alpha K_B} ):( P_A = frac{r_B}{beta} left(1 - frac{r_A}{alpha K_B} + frac{r_A}{alpha K_B} cdot frac{P_A}{K_A} right) )Let me denote ( d = 1 - frac{r_A}{alpha K_B} ) and ( e = frac{r_A}{alpha K_B K_A} ). So, the equation becomes:( P_A = frac{r_B}{beta} (d + e P_A) )Expanding:( P_A = frac{r_B d}{beta} + frac{r_B e}{beta} P_A )Bring the term with ( P_A ) to the left:( P_A - frac{r_B e}{beta} P_A = frac{r_B d}{beta} )Factor out ( P_A ):( P_A left(1 - frac{r_B e}{beta}right) = frac{r_B d}{beta} )Solve for ( P_A ):( P_A = frac{frac{r_B d}{beta}}{1 - frac{r_B e}{beta}} )Simplify numerator and denominator:( P_A = frac{r_B d}{beta - r_B e} )Now, substitute back ( d = 1 - frac{r_A}{alpha K_B} ) and ( e = frac{r_A}{alpha K_B K_A} ):( P_A = frac{r_B left(1 - frac{r_A}{alpha K_B}right)}{beta - r_B cdot frac{r_A}{alpha K_B K_A}} )Let me factor out ( frac{r_A}{alpha K_B} ) in the denominator:Denominator: ( beta - frac{r_A r_B}{alpha K_B K_A} )So, ( P_A = frac{r_B left(1 - frac{r_A}{alpha K_B}right)}{beta - frac{r_A r_B}{alpha K_B K_A}} )Similarly, once we have ( P_A ), we can find ( P_B ) using equation (1a):( P_B = frac{r_A}{alpha} left(1 - frac{P_A}{K_A}right) )So, that's the non-trivial equilibrium point.Now, to assess the stability, I need to linearize the system around the equilibrium points and analyze the eigenvalues of the Jacobian matrix.The Jacobian matrix ( J ) is given by:[J = begin{bmatrix}frac{partial}{partial P_A} left( r_A P_A (1 - P_A / K_A) - alpha P_A P_B right) & frac{partial}{partial P_B} left( r_A P_A (1 - P_A / K_A) - alpha P_A P_B right) frac{partial}{partial P_A} left( r_B P_B (1 - P_B / K_B) - beta P_A P_B right) & frac{partial}{partial P_B} left( r_B P_B (1 - P_B / K_B) - beta P_A P_B right)end{bmatrix}]Calculating each partial derivative:First, for ( frac{partial}{partial P_A} ) of the first equation:( frac{d}{dP_A} [ r_A P_A (1 - P_A / K_A) - alpha P_A P_B ] = r_A (1 - P_A / K_A) - r_A P_A / K_A - alpha P_B )Simplify:( r_A (1 - 2 P_A / K_A) - alpha P_B )Similarly, ( frac{partial}{partial P_B} ) of the first equation:( frac{d}{dP_B} [ r_A P_A (1 - P_A / K_A) - alpha P_A P_B ] = - alpha P_A )For the second equation, ( frac{partial}{partial P_A} ):( frac{d}{dP_A} [ r_B P_B (1 - P_B / K_B) - beta P_A P_B ] = - beta P_B )And ( frac{partial}{partial P_B} ):( frac{d}{dP_B} [ r_B P_B (1 - P_B / K_B) - beta P_A P_B ] = r_B (1 - 2 P_B / K_B) - beta P_A )So, putting it all together, the Jacobian matrix at equilibrium is:[J = begin{bmatrix}r_A (1 - 2 P_A / K_A) - alpha P_B & - alpha P_A - beta P_B & r_B (1 - 2 P_B / K_B) - beta P_Aend{bmatrix}]Now, to assess stability, we need to evaluate the Jacobian at each equilibrium point and find the eigenvalues.First, consider the trivial equilibrium ( (0, 0) ).At ( (0, 0) ):[J = begin{bmatrix}r_A & 0 0 & r_Bend{bmatrix}]The eigenvalues are ( r_A ) and ( r_B ). Since ( r_A ) and ( r_B ) are intrinsic growth rates, they are positive. Therefore, the trivial equilibrium is unstable (a source).Next, consider the non-trivial equilibrium ( (P_A^*, P_B^*) ). We need to compute the Jacobian at this point.Let me denote ( P_A^* ) and ( P_B^* ) as the equilibrium values.From the equilibrium conditions:1. ( r_A (1 - P_A^* / K_A) = alpha P_B^* )2. ( r_B (1 - P_B^* / K_B) = beta P_A^* )So, let's substitute these into the Jacobian.First, the (1,1) entry:( r_A (1 - 2 P_A^* / K_A) - alpha P_B^* )But from equation 1, ( r_A (1 - P_A^* / K_A) = alpha P_B^* ). So,( r_A (1 - 2 P_A^* / K_A) = r_A (1 - P_A^* / K_A) - r_A P_A^* / K_A = alpha P_B^* - r_A P_A^* / K_A )Therefore, the (1,1) entry becomes:( alpha P_B^* - r_A P_A^* / K_A - alpha P_B^* = - r_A P_A^* / K_A )Similarly, the (2,2) entry:( r_B (1 - 2 P_B^* / K_B) - beta P_A^* )From equation 2, ( r_B (1 - P_B^* / K_B) = beta P_A^* ). So,( r_B (1 - 2 P_B^* / K_B) = r_B (1 - P_B^* / K_B) - r_B P_B^* / K_B = beta P_A^* - r_B P_B^* / K_B )Therefore, the (2,2) entry becomes:( beta P_A^* - r_B P_B^* / K_B - beta P_A^* = - r_B P_B^* / K_B )So, the Jacobian at the non-trivial equilibrium simplifies to:[J = begin{bmatrix}- frac{r_A P_A^*}{K_A} & - alpha P_A^* - beta P_B^* & - frac{r_B P_B^*}{K_B}end{bmatrix}]Now, to find the eigenvalues, we need to solve the characteristic equation:[det(J - lambda I) = 0]Which is:[left( - frac{r_A P_A^*}{K_A} - lambda right) left( - frac{r_B P_B^*}{K_B} - lambda right) - ( alpha P_A^* beta P_B^* ) = 0]Expanding the determinant:[left( frac{r_A P_A^*}{K_A} + lambda right) left( frac{r_B P_B^*}{K_B} + lambda right) - alpha beta P_A^* P_B^* = 0]Let me denote ( a = frac{r_A P_A^*}{K_A} ) and ( b = frac{r_B P_B^*}{K_B} ). Then, the equation becomes:[(a + lambda)(b + lambda) - alpha beta P_A^* P_B^* = 0]Expanding:( ab + a lambda + b lambda + lambda^2 - alpha beta P_A^* P_B^* = 0 )So,( lambda^2 + (a + b) lambda + (ab - alpha beta P_A^* P_B^*) = 0 )To find the eigenvalues, we can use the quadratic formula:( lambda = frac{ - (a + b) pm sqrt{(a + b)^2 - 4 (ab - alpha beta P_A^* P_B^*)} }{2} )Simplify the discriminant:( D = (a + b)^2 - 4 (ab - alpha beta P_A^* P_B^*) = a^2 + 2ab + b^2 - 4ab + 4 alpha beta P_A^* P_B^* )( D = a^2 - 2ab + b^2 + 4 alpha beta P_A^* P_B^* = (a - b)^2 + 4 alpha beta P_A^* P_B^* )Since ( (a - b)^2 ) is always non-negative and ( alpha, beta, P_A^*, P_B^* ) are positive constants, the discriminant ( D ) is positive. Therefore, we have two real eigenvalues.The eigenvalues are:( lambda = frac{ - (a + b) pm sqrt{(a - b)^2 + 4 alpha beta P_A^* P_B^*} }{2} )Now, the sign of the eigenvalues determines the stability. If both eigenvalues are negative, the equilibrium is stable (a sink). If at least one eigenvalue is positive, it's unstable.Given that ( a = frac{r_A P_A^*}{K_A} > 0 ) and ( b = frac{r_B P_B^*}{K_B} > 0 ), the sum ( a + b > 0 ), so the real parts of the eigenvalues are negative if the numerator is negative. However, since the discriminant is positive and larger than ( (a + b)^2 ), the square root term is larger than ( a + b ), making the numerator negative. Therefore, both eigenvalues have negative real parts, meaning the equilibrium is stable.Wait, hold on. Let me think again. The eigenvalues are:( lambda = frac{ - (a + b) pm sqrt{(a - b)^2 + 4 alpha beta P_A^* P_B^*} }{2} )So, the two eigenvalues are:1. ( lambda_1 = frac{ - (a + b) + sqrt{(a - b)^2 + 4 alpha beta P_A^* P_B^*} }{2} )2. ( lambda_2 = frac{ - (a + b) - sqrt{(a - b)^2 + 4 alpha beta P_A^* P_B^*} }{2} )Since ( sqrt{(a - b)^2 + 4 alpha beta P_A^* P_B^*} geq |a - b| ), the first eigenvalue ( lambda_1 ) could be positive or negative depending on whether ( sqrt{(a - b)^2 + 4 alpha beta P_A^* P_B^*} > a + b ).Wait, let's compute:( sqrt{(a - b)^2 + 4 alpha beta P_A^* P_B^*} )Compare it to ( a + b ):Is ( (a - b)^2 + 4 alpha beta P_A^* P_B^* > (a + b)^2 )?Compute the difference:( (a - b)^2 + 4 alpha beta P_A^* P_B^* - (a + b)^2 = (a^2 - 2ab + b^2) + 4 alpha beta P_A^* P_B^* - (a^2 + 2ab + b^2) = -4ab + 4 alpha beta P_A^* P_B^* )So, the difference is ( 4 ( alpha beta P_A^* P_B^* - ab ) )Recall that ( a = frac{r_A P_A^*}{K_A} ) and ( b = frac{r_B P_B^*}{K_B} ). So,( ab = frac{r_A r_B P_A^* P_B^*}{K_A K_B} )Therefore, the difference is:( 4 ( alpha beta P_A^* P_B^* - frac{r_A r_B P_A^* P_B^*}{K_A K_B} ) = 4 P_A^* P_B^* ( alpha beta - frac{r_A r_B}{K_A K_B} ) )So, if ( alpha beta > frac{r_A r_B}{K_A K_B} ), then the difference is positive, meaning ( sqrt{(a - b)^2 + 4 alpha beta P_A^* P_B^*} > a + b ), so ( lambda_1 ) becomes positive.If ( alpha beta < frac{r_A r_B}{K_A K_B} ), then the difference is negative, so ( sqrt{(a - b)^2 + 4 alpha beta P_A^* P_B^*} < a + b ), making ( lambda_1 ) negative.Therefore, the stability depends on the sign of ( alpha beta - frac{r_A r_B}{K_A K_B} ).If ( alpha beta > frac{r_A r_B}{K_A K_B} ), then one eigenvalue is positive, and the other is negative (since ( lambda_2 ) is always negative because the numerator is negative). Therefore, the equilibrium is a saddle point, which is unstable.If ( alpha beta < frac{r_A r_B}{K_A K_B} ), then both eigenvalues are negative, making the equilibrium stable.If ( alpha beta = frac{r_A r_B}{K_A K_B} ), then the discriminant becomes ( (a - b)^2 ), so the eigenvalues are ( lambda = frac{ - (a + b) pm |a - b| }{2} ). Depending on whether ( a > b ) or ( b > a ), one eigenvalue is zero and the other is negative, making it a line of equilibria or a node with one zero eigenvalue.Therefore, the non-trivial equilibrium is stable if ( alpha beta < frac{r_A r_B}{K_A K_B} ), unstable if ( alpha beta > frac{r_A r_B}{K_A K_B} ), and has a line of equilibria if they are equal.So, summarizing the equilibrium points:1. Trivial equilibrium ( (0, 0) ): Unstable.2. Non-trivial equilibrium ( (P_A^*, P_B^*) ): Stable if ( alpha beta < frac{r_A r_B}{K_A K_B} ), unstable otherwise.Part 2: Biochemical Compound ExtractionDr. Elena extracts compounds from both plants. The concentrations are given by:( C_A(t) = C_0 e^{-lambda t} )( C_B(t) = C_0 e^{-mu t} )She wants to find the time ( t ) that maximizes the total concentration ( C_A(t) + C_B(t) ). Additionally, if the active ingredients combine synergistically according to ( S(t) = k C_A(t) C_B(t) ), find the time ( t ) that maximizes ( S(t) ).First, let's tackle the total concentration.Maximizing Total Concentration ( C_A(t) + C_B(t) )We need to maximize ( C(t) = C_A(t) + C_B(t) = C_0 e^{-lambda t} + C_0 e^{-mu t} )To find the maximum, take the derivative with respect to ( t ) and set it to zero.Compute ( C'(t) ):( C'(t) = - lambda C_0 e^{-lambda t} - mu C_0 e^{-mu t} )Set ( C'(t) = 0 ):( - lambda C_0 e^{-lambda t} - mu C_0 e^{-mu t} = 0 )Divide both sides by ( -C_0 ):( lambda e^{-lambda t} + mu e^{-mu t} = 0 )But ( e^{-lambda t} ) and ( e^{-mu t} ) are always positive, and ( lambda ), ( mu ) are positive constants (decay rates). Therefore, the sum of positive terms cannot be zero. This implies that ( C'(t) ) is always negative, meaning ( C(t) ) is a strictly decreasing function.Therefore, the maximum occurs at ( t = 0 ).Wait, that seems counterintuitive. Let me double-check.Given that both ( C_A(t) ) and ( C_B(t) ) are decreasing functions (since ( lambda, mu > 0 )), their sum is also decreasing. Therefore, the maximum is indeed at ( t = 0 ).But let me think again. Maybe the problem is considering the time after extraction, so perhaps the concentrations are being extracted over time, but the functions given are exponential decays. So, yes, the concentrations are highest at ( t = 0 ).So, the time that maximizes the total concentration is ( t = 0 ).Maximizing Synergistic Function ( S(t) = k C_A(t) C_B(t) )Now, ( S(t) = k C_A(t) C_B(t) = k C_0^2 e^{-(lambda + mu) t} )Wait, that's just an exponential decay as well. So, similar to the previous case, ( S(t) ) is decreasing since the exponent is negative. Therefore, the maximum occurs at ( t = 0 ).But wait, let me compute it properly.Compute ( S(t) = k C_0 e^{-lambda t} cdot C_0 e^{-mu t} = k C_0^2 e^{-(lambda + mu) t} )Yes, so ( S(t) ) is an exponential decay function. Therefore, its maximum is at ( t = 0 ).But hold on, maybe I misread the problem. Is the function ( S(t) = k C_A(t) C_B(t) ) or is it something else? The problem says "synergistic according to the function ( S(t) = k C_A(t) C_B(t) )", so yes, it's the product.Therefore, since both ( C_A(t) ) and ( C_B(t) ) are decreasing, their product is also decreasing, so maximum at ( t = 0 ).Wait, but maybe if the decay rates are different, the product could have a maximum somewhere else? Let me check.Suppose ( lambda neq mu ). Let me compute the derivative of ( S(t) ):( S(t) = k C_0^2 e^{-(lambda + mu) t} )Therefore, ( S'(t) = -k C_0^2 (lambda + mu) e^{-(lambda + mu) t} )Set ( S'(t) = 0 ):( -k C_0^2 (lambda + mu) e^{-(lambda + mu) t} = 0 )But ( e^{-(lambda + mu) t} ) is never zero, so there's no solution. Therefore, ( S(t) ) is always decreasing, and the maximum is at ( t = 0 ).Alternatively, if the function was ( S(t) = k C_A(t) C_B(t) ) without the exponential, but no, it's still the same.Wait, unless the concentrations are increasing? But no, the functions are exponential decays.Therefore, both the total concentration and the synergistic function are maximized at ( t = 0 ).But let me think again. Maybe the problem is considering the time after extraction, so perhaps the concentrations are being extracted over time, but the functions given are exponential decays. So, yes, the concentrations are highest at ( t = 0 ).Alternatively, if the concentrations were increasing, but the given functions are decays, so no.Therefore, the conclusion is that both maxima occur at ( t = 0 ).But wait, maybe I'm missing something. Let me think about the total concentration again.Total concentration is ( C_A(t) + C_B(t) ). If ( lambda neq mu ), could the sum have a maximum somewhere else? Let's see.Suppose ( lambda neq mu ). Let me compute the derivative again:( C'(t) = - lambda C_0 e^{-lambda t} - mu C_0 e^{-mu t} )Set to zero:( lambda e^{-lambda t} + mu e^{-mu t} = 0 )But since exponentials are positive, and ( lambda, mu > 0 ), the left side is always positive. Therefore, no solution. So, ( C(t) ) is always decreasing, maximum at ( t = 0 ).Similarly, for ( S(t) ), the derivative is always negative, so maximum at ( t = 0 ).Therefore, both maxima occur at ( t = 0 ).But wait, in reality, if you have two decaying exponentials, their sum might have a maximum at some point if one is decaying faster than the other. Wait, no, because both are decaying, their sum is always decreasing.Wait, let me plot it mentally. Suppose ( lambda = 1 ), ( mu = 2 ). Then, ( C_A(t) = e^{-t} ), ( C_B(t) = e^{-2t} ). The sum is ( e^{-t} + e^{-2t} ). The derivative is ( -e^{-t} - 2 e^{-2t} ), which is always negative. So, the sum is always decreasing.Therefore, yes, the maximum is at ( t = 0 ).So, both the total concentration and the synergistic function are maximized at ( t = 0 ).But let me check the problem statement again. It says "find the time ( t ) at which the total concentration... is maximized" and "determine the time ( t ) that maximizes ( S(t) )". So, the answer is ( t = 0 ) for both.But wait, in some cases, if the functions were not decaying, but say, logistic growth or something, the maximum could be somewhere else. But in this case, they are exponential decays, so no.Therefore, the answer is ( t = 0 ) for both.Final Answer1. The equilibrium points are ( (0, 0) ) (unstable) and ( (P_A^*, P_B^*) ) (stable if ( alpha beta < frac{r_A r_B}{K_A K_B} ), unstable otherwise). 2. The time that maximizes the total concentration and the synergistic function is ( boxed{0} ).</think>

🧠:<think>Okay, so I have this problem about modeling recidivism rates using a logistic growth model. It's divided into two sub-problems. Let me try to tackle them step by step.Starting with Sub-problem 1: I need to find the time ( t ) when the recidivism rate ( R(t) ) reaches 50% of the maximum rate ( L ). The given logistic model is:[ R(t) = frac{L}{1 + e^{-k(t - t_0)}} ]They provided the values: ( L = 0.6 ), ( k = 0.5 ), and ( t_0 = 2 ) years. So, I need to find ( t ) when ( R(t) = 0.5 times L ). Since ( L = 0.6 ), 50% of that would be ( 0.3 ). So, setting up the equation:[ 0.3 = frac{0.6}{1 + e^{-0.5(t - 2)}} ]Hmm, okay. Let me write that down:[ 0.3 = frac{0.6}{1 + e^{-0.5(t - 2)}} ]I need to solve for ( t ). Let me rearrange this equation. First, I can multiply both sides by the denominator to get rid of the fraction:[ 0.3 times (1 + e^{-0.5(t - 2)}) = 0.6 ]Divide both sides by 0.3 to simplify:[ 1 + e^{-0.5(t - 2)} = 2 ]Subtract 1 from both sides:[ e^{-0.5(t - 2)} = 1 ]Wait, ( e^0 = 1 ), so that would mean:[ -0.5(t - 2) = 0 ]Solving for ( t ):[ -0.5t + 1 = 0 ][ -0.5t = -1 ][ t = 2 ]Wait, that seems straightforward. So, at ( t = 2 ) years, the recidivism rate is 50% of the maximum. But ( t_0 ) is also 2 years. I remember that in logistic growth models, ( t_0 ) is the inflection point where the growth rate is the highest. So, it makes sense that at ( t_0 ), the function reaches half of its maximum value. That checks out.So, Sub-problem 1 answer is ( t = 2 ) years.Moving on to Sub-problem 2: They say that implementing a new program reduces the growth rate constant ( k ) by 20%. So, the original ( k ) was 0.5, reducing it by 20% would make the new ( k' = 0.5 - 0.2 times 0.5 = 0.5 - 0.1 = 0.4 ).So, the new growth rate constant is 0.4. Now, I need to find the new time ( t' ) when the recidivism rate reaches 50% of ( L ) again. Using the same model:[ R(t) = frac{0.6}{1 + e^{-0.4(t - 2)}} ]Again, setting ( R(t) = 0.3 ):[ 0.3 = frac{0.6}{1 + e^{-0.4(t' - 2)}} ]Same steps as before. Multiply both sides by denominator:[ 0.3 times (1 + e^{-0.4(t' - 2)}) = 0.6 ]Divide both sides by 0.3:[ 1 + e^{-0.4(t' - 2)} = 2 ]Subtract 1:[ e^{-0.4(t' - 2)} = 1 ]Again, ( e^0 = 1 ), so:[ -0.4(t' - 2) = 0 ]Solving for ( t' ):[ -0.4t' + 0.8 = 0 ][ -0.4t' = -0.8 ][ t' = 2 ]Wait, that's the same result as before. So, even with a reduced growth rate, ( t' ) is still 2 years. Hmm, that seems a bit counterintuitive. I thought that reducing the growth rate would change the time when the rate reaches 50%.But let me think again. The logistic function's inflection point is at ( t_0 ), which is when the function is at half of its maximum. So, regardless of the growth rate ( k ), the inflection point ( t_0 ) is fixed. Therefore, even if ( k ) changes, the time when the function reaches 50% of ( L ) is still at ( t_0 ).But wait, in the logistic model, ( t_0 ) is the time at which the growth rate is maximum, which is also where the function is at half its maximum. So, if ( t_0 ) is fixed at 2 years, then regardless of ( k ), the time when ( R(t) = 0.5L ) is still 2 years.But hold on, in the logistic model, is ( t_0 ) dependent on ( k )? Let me recall the standard logistic function:[ R(t) = frac{L}{1 + e^{-k(t - t_0)}} ]Here, ( t_0 ) is the time when the function is at half of ( L ), and it's independent of ( k ). So, if ( k ) changes, the steepness of the curve changes, but the inflection point ( t_0 ) remains the same.Therefore, even with a different ( k ), as long as ( t_0 ) is fixed, the time when ( R(t) = 0.5L ) is still ( t_0 ).But in this problem, is ( t_0 ) fixed? Let me check the original problem statement.It says: "the recidivism rate follows a logistic growth model given by... where ( t_0 ) is the time at which the rate of recidivism is increasing most rapidly."So, in the model, ( t_0 ) is fixed at 2 years. So, even if ( k ) changes, ( t_0 ) remains 2 years. Therefore, the time when the recidivism rate reaches 50% of ( L ) is still 2 years, regardless of ( k ).But wait, that seems contradictory because if ( k ) is smaller, the curve is less steep, so it might take longer to reach 50%? Or does it?Wait, actually, in the logistic function, ( t_0 ) is the point where the function is at half its maximum, regardless of ( k ). So, even if ( k ) is smaller, the function will still cross 50% at ( t_0 ). The difference is that with a smaller ( k ), the function approaches the maximum more slowly after ( t_0 ), but it still reaches 50% at ( t_0 ).So, in this case, even after reducing ( k ) by 20%, the time ( t' ) when ( R(t) = 0.3 ) is still 2 years.Therefore, both Sub-problem 1 and Sub-problem 2 result in ( t = 2 ) years.But wait, that seems odd because I thought changing ( k ) would affect the time. Maybe I need to think about this differently.Alternatively, perhaps I misinterpreted the problem. Maybe ( t_0 ) is not fixed, but is a function of ( k ). Let me recall the standard logistic function.Wait, in the standard logistic function, ( t_0 ) is the time of the inflection point, which is when the growth rate is maximum. So, if ( k ) changes, does ( t_0 ) change? Or is ( t_0 ) a parameter that can be set independently?In the given problem, ( t_0 ) is given as 2 years, regardless of ( k ). So, even if ( k ) changes, ( t_0 ) remains 2 years. Therefore, the time when the function is at 50% is still 2 years.But let me verify this by plugging in the numbers.Original model: ( R(t) = frac{0.6}{1 + e^{-0.5(t - 2)}} ). At ( t = 2 ), ( R(2) = 0.6 / (1 + e^0) = 0.6 / 2 = 0.3 ). So, 50% of L is achieved at t=2.After reducing k to 0.4, the model becomes ( R(t) = frac{0.6}{1 + e^{-0.4(t - 2)}} ). At ( t = 2 ), ( R(2) = 0.6 / (1 + e^0) = 0.3 ). So, same result.Therefore, regardless of ( k ), as long as ( t_0 ) is fixed, the time when ( R(t) = 0.5L ) is ( t_0 ).So, in both cases, the time is 2 years.But then, what's the implication? If the growth rate ( k ) is reduced, the curve becomes less steep, meaning it approaches the maximum more slowly after the inflection point. So, before ( t_0 ), the recidivism rate increases, but with a smaller ( k ), it would have been increasing more slowly. After ( t_0 ), it continues to increase but at a decreasing rate.Wait, but in terms of reaching 50%, it's still at ( t_0 ). So, the implication is that the timing of when the recidivism rate reaches half the maximum doesn't change, but the rate at which it approaches the maximum changes.Therefore, implementing a program that reduces ( k ) would mean that after the inflection point, the recidivism rate doesn't increase as rapidly towards the maximum. So, the peak recidivism rate is still 0.6, but it takes longer to reach higher rates beyond 50%.So, for parole programs, this implies that reducing ( k ) (the growth rate) would lead to a slower increase in recidivism rates after the initial peak. This could be beneficial because it might give more time for interventions or support programs to help reduce recidivism before it reaches higher levels.Alternatively, it might mean that the recidivism rate doesn't spike as quickly, which could be advantageous for managing prison populations or allocating resources.But in terms of the specific question, the time when the recidivism rate reaches 50% remains the same, but the dynamics after that point change.Wait, but let me think again. If ( k ) is smaller, does the function take longer to reach 50%? Or is it still at ( t_0 )?From the calculations, it's still at ( t_0 ). So, the timing is the same, but the shape of the curve changes. So, the rate at which it approaches 50% is different, but the crossing point is the same.Therefore, the implication is that the timing of when the recidivism rate reaches 50% doesn't change, but the slope at that point changes. With a smaller ( k ), the slope is less steep, meaning the rate of increase in recidivism is slower after that point.So, for parole programs, this suggests that while the initial peak at 50% occurs at the same time, the subsequent rise towards the maximum rate is more gradual. This could be beneficial as it might allow more time for interventions or for the programs to have a lasting effect before recidivism rates climb higher.Alternatively, it might indicate that the programs are more effective in the long term, slowing down the progression towards higher recidivism rates.But in terms of the specific question, the time ( t' ) is still 2 years, same as before. So, the answer is that ( t' = 2 ) years as well, and the implication is that while the timing remains the same, the rate of increase after that point is slower, which could be advantageous for managing recidivism.Wait, but I'm a bit confused because I thought changing ( k ) would affect the time to reach 50%. Maybe I need to double-check the math.Let me re-examine the equation for Sub-problem 2:[ 0.3 = frac{0.6}{1 + e^{-0.4(t' - 2)}} ]Multiply both sides by denominator:[ 0.3(1 + e^{-0.4(t' - 2)}) = 0.6 ]Divide by 0.3:[ 1 + e^{-0.4(t' - 2)} = 2 ]Subtract 1:[ e^{-0.4(t' - 2)} = 1 ]Take natural log:[ -0.4(t' - 2) = 0 ]So,[ t' - 2 = 0 ][ t' = 2 ]Yes, same result. So, regardless of ( k ), as long as ( t_0 ) is fixed, the time when ( R(t) = 0.5L ) is ( t_0 ).Therefore, the new time ( t' ) is still 2 years. The implication is that while the growth rate ( k ) affects how quickly the recidivism rate approaches the maximum after ( t_0 ), the time when it reaches 50% remains unchanged.So, for parole programs, this suggests that implementing a program that reduces ( k ) doesn't change when the recidivism rate reaches 50%, but it does make the approach to the maximum rate slower. This could be beneficial as it might give more time for interventions or for the effects of the program to take hold before recidivism rates escalate further.Alternatively, it might mean that the program is more effective in the long term, preventing a rapid increase in recidivism after the initial peak.In summary, both Sub-problems result in ( t = 2 ) years, but the reduced ( k ) leads to a slower growth rate after that point, which could be advantageous for managing recidivism.</think>

🧠:<think>Alright, so I have this problem where a freelance writer needs to evaluate a bar's new cocktail menu. The bar has 8 distinct cocktails, each with unique ingredient combinations. The writer has to provide feedback on all possible pairings to analyze complementary flavors. The first part is about combinatorics and probability, asking for the total number of unique cocktail pairings. The second part is more complex, involving optimization and linear algebra. It requires finding the pair of cocktails with the smallest Euclidean distance between their flavor vectors in a 5-dimensional space. Each vector represents the intensity of five flavors: sweetness, sourness, bitterness, saltiness, and umami.Starting with the first question: calculating the total number of unique pairings. I remember that when you want to find the number of ways to choose 2 items out of a set without considering the order, you use combinations. The formula for combinations is n choose k, which is n! / (k! * (n - k)!). Here, n is 8 because there are 8 cocktails, and k is 2 because we're pairing them two at a time.So, plugging in the numbers: 8 choose 2 is 8! / (2! * (8 - 2)!) = (8 * 7 * 6!) / (2 * 1 * 6!) = (8 * 7) / 2 = 56 / 2 = 28. Therefore, there are 28 unique pairings. That seems straightforward.Moving on to the second part: finding the pair of cocktails with the smallest Euclidean distance. Each cocktail is represented by a 5-dimensional vector. The Euclidean distance between two vectors is calculated by taking the square root of the sum of the squared differences of their corresponding components.To find the pair with the smallest distance, I need to compute the distance between every possible pair of cocktails and then identify the minimum. Since there are 8 cocktails, that means I have to compute 28 distances. That's a bit tedious, but manageable.Let me list out all the cocktails with their vectors for clarity:C1: (3, 7, 2, 5, 4)C2: (6, 1, 3, 2, 8)C3: (4, 5, 6, 1, 3)C4: (7, 2, 1, 8, 6)C5: (2, 4, 7, 3, 5)C6: (8, 3, 5, 6, 2)C7: (1, 6, 4, 7, 3)C8: (5, 2, 8, 4, 7)I think the best approach is to compute the distance between each pair systematically. Maybe I can start with C1 and compute its distance with C2, C3, ..., C8, then move on to C2 and compute its distance with C3, C4, ..., C8, and so on. That way, I cover all 28 pairs without repetition.Let me recall the Euclidean distance formula:Distance between Ci and Cj = sqrt[(x1 - x2)^2 + (y1 - y2)^2 + (z1 - z2)^2 + ...] for each dimension.But since we're looking for the smallest distance, maybe we can just compare the squared distances to avoid computing square roots, which can save some computation time. The pair with the smallest squared distance will also have the smallest actual distance.So, let's proceed step by step.First, compute distances starting with C1:1. C1 & C2:Compute the squared differences for each dimension:(3-6)^2 = 9(7-1)^2 = 36(2-3)^2 = 1(5-2)^2 = 9(4-8)^2 = 16Sum: 9 + 36 + 1 + 9 + 16 = 712. C1 & C3:(3-4)^2 = 1(7-5)^2 = 4(2-6)^2 = 16(5-1)^2 = 16(4-3)^2 = 1Sum: 1 + 4 + 16 + 16 + 1 = 383. C1 & C4:(3-7)^2 = 16(7-2)^2 = 25(2-1)^2 = 1(5-8)^2 = 9(4-6)^2 = 4Sum: 16 + 25 + 1 + 9 + 4 = 554. C1 & C5:(3-2)^2 = 1(7-4)^2 = 9(2-7)^2 = 25(5-3)^2 = 4(4-5)^2 = 1Sum: 1 + 9 + 25 + 4 + 1 = 405. C1 & C6:(3-8)^2 = 25(7-3)^2 = 16(2-5)^2 = 9(5-6)^2 = 1(4-2)^2 = 4Sum: 25 + 16 + 9 + 1 + 4 = 556. C1 & C7:(3-1)^2 = 4(7-6)^2 = 1(2-4)^2 = 4(5-7)^2 = 4(4-3)^2 = 1Sum: 4 + 1 + 4 + 4 + 1 = 147. C1 & C8:(3-5)^2 = 4(7-2)^2 = 25(2-8)^2 = 36(5-4)^2 = 1(4-7)^2 = 9Sum: 4 + 25 + 36 + 1 + 9 = 75So, the squared distances for C1 with others are: 71, 38, 55, 40, 55, 14, 75.The smallest here is 14 between C1 and C7.Now, moving on to C2:1. C2 & C3:(6-4)^2 = 4(1-5)^2 = 16(3-6)^2 = 9(2-1)^2 = 1(8-3)^2 = 25Sum: 4 + 16 + 9 + 1 + 25 = 552. C2 & C4:(6-7)^2 = 1(1-2)^2 = 1(3-1)^2 = 4(2-8)^2 = 36(8-6)^2 = 4Sum: 1 + 1 + 4 + 36 + 4 = 463. C2 & C5:(6-2)^2 = 16(1-4)^2 = 9(3-7)^2 = 16(2-3)^2 = 1(8-5)^2 = 9Sum: 16 + 9 + 16 + 1 + 9 = 514. C2 & C6:(6-8)^2 = 4(1-3)^2 = 4(3-5)^2 = 4(2-6)^2 = 16(8-2)^2 = 36Sum: 4 + 4 + 4 + 16 + 36 = 645. C2 & C7:(6-1)^2 = 25(1-6)^2 = 25(3-4)^2 = 1(2-7)^2 = 25(8-3)^2 = 25Sum: 25 + 25 + 1 + 25 + 25 = 1016. C2 & C8:(6-5)^2 = 1(1-2)^2 = 1(3-8)^2 = 25(2-4)^2 = 4(8-7)^2 = 1Sum: 1 + 1 + 25 + 4 + 1 = 32So, squared distances for C2: 55, 46, 51, 64, 101, 32.The smallest here is 32 between C2 and C8.Next, C3:1. C3 & C4:(4-7)^2 = 9(5-2)^2 = 9(6-1)^2 = 25(1-8)^2 = 49(3-6)^2 = 9Sum: 9 + 9 + 25 + 49 + 9 = 1012. C3 & C5:(4-2)^2 = 4(5-4)^2 = 1(6-7)^2 = 1(1-3)^2 = 4(3-5)^2 = 4Sum: 4 + 1 + 1 + 4 + 4 = 143. C3 & C6:(4-8)^2 = 16(5-3)^2 = 4(6-5)^2 = 1(1-6)^2 = 25(3-2)^2 = 1Sum: 16 + 4 + 1 + 25 + 1 = 474. C3 & C7:(4-1)^2 = 9(5-6)^2 = 1(6-4)^2 = 4(1-7)^2 = 36(3-3)^2 = 0Sum: 9 + 1 + 4 + 36 + 0 = 505. C3 & C8:(4-5)^2 = 1(5-2)^2 = 9(6-8)^2 = 4(1-4)^2 = 9(3-7)^2 = 16Sum: 1 + 9 + 4 + 9 + 16 = 39So, squared distances for C3: 101, 14, 47, 50, 39.The smallest here is 14 between C3 and C5.Moving on to C4:1. C4 & C5:(7-2)^2 = 25(2-4)^2 = 4(1-7)^2 = 36(8-3)^2 = 25(6-5)^2 = 1Sum: 25 + 4 + 36 + 25 + 1 = 912. C4 & C6:(7-8)^2 = 1(2-3)^2 = 1(1-5)^2 = 16(8-6)^2 = 4(6-2)^2 = 16Sum: 1 + 1 + 16 + 4 + 16 = 383. C4 & C7:(7-1)^2 = 36(2-6)^2 = 16(1-4)^2 = 9(8-7)^2 = 1(6-3)^2 = 9Sum: 36 + 16 + 9 + 1 + 9 = 714. C4 & C8:(7-5)^2 = 4(2-2)^2 = 0(1-8)^2 = 49(8-4)^2 = 16(6-7)^2 = 1Sum: 4 + 0 + 49 + 16 + 1 = 70So, squared distances for C4: 91, 38, 71, 70.The smallest here is 38 between C4 and C6.Next, C5:1. C5 & C6:(2-8)^2 = 36(4-3)^2 = 1(7-5)^2 = 4(3-6)^2 = 9(5-2)^2 = 9Sum: 36 + 1 + 4 + 9 + 9 = 592. C5 & C7:(2-1)^2 = 1(4-6)^2 = 4(7-4)^2 = 9(3-7)^2 = 16(5-3)^2 = 4Sum: 1 + 4 + 9 + 16 + 4 = 343. C5 & C8:(2-5)^2 = 9(4-2)^2 = 4(7-8)^2 = 1(3-4)^2 = 1(5-7)^2 = 4Sum: 9 + 4 + 1 + 1 + 4 = 19So, squared distances for C5: 59, 34, 19.The smallest here is 19 between C5 and C8.Moving on to C6:1. C6 & C7:(8-1)^2 = 49(3-6)^2 = 9(5-4)^2 = 1(6-7)^2 = 1(2-3)^2 = 1Sum: 49 + 9 + 1 + 1 + 1 = 612. C6 & C8:(8-5)^2 = 9(3-2)^2 = 1(5-8)^2 = 9(6-4)^2 = 4(2-7)^2 = 25Sum: 9 + 1 + 9 + 4 + 25 = 48So, squared distances for C6: 61, 48.The smallest here is 48 between C6 and C8.Finally, C7:1. C7 & C8:(1-5)^2 = 16(6-2)^2 = 16(4-8)^2 = 16(7-4)^2 = 9(3-7)^2 = 16Sum: 16 + 16 + 16 + 9 + 16 = 73So, squared distance for C7 is 73.Now, compiling all the smallest squared distances from each set:- C1: 14 (with C7)- C2: 32 (with C8)- C3: 14 (with C5)- C4: 38 (with C6)- C5: 19 (with C8)- C6: 48 (with C8)- C7: 73 (with C8)Looking at these, the smallest squared distance is 14, which occurs between C1 & C7 and between C3 & C5. So, both pairs have the same squared distance of 14. Wait, so both pairs have the same minimal distance? That means there are two pairs with the smallest Euclidean distance. Let me confirm their distances:For C1 & C7: sqrt(14) ≈ 3.7417For C3 & C5: sqrt(14) ≈ 3.7417So, both pairs have the same minimal distance. Therefore, the writer should consider both pairs as having the most similar flavor profiles.But the question says "the pair of cocktails", implying a single pair. Hmm, maybe I made a mistake because the squared distances are the same, but perhaps the actual distances are the same as well. So, both pairs are equally similar.Alternatively, perhaps I miscalculated somewhere. Let me double-check the distances for C1 & C7 and C3 & C5.C1: (3,7,2,5,4)C7: (1,6,4,7,3)Differences:3-1=2, squared=47-6=1, squared=12-4=-2, squared=45-7=-2, squared=44-3=1, squared=1Total: 4+1+4+4+1=14. Correct.C3: (4,5,6,1,3)C5: (2,4,7,3,5)Differences:4-2=2, squared=45-4=1, squared=16-7=-1, squared=11-3=-2, squared=43-5=-2, squared=4Total: 4+1+1+4+4=14. Correct.So, both pairs indeed have the same squared distance. Therefore, both are equally the closest pair. But the question asks for "the pair of cocktails", which might suggest a single answer. Maybe I need to present both pairs as the answer. Alternatively, perhaps I missed a closer pair.Wait, let me check all the squared distances again to ensure I didn't miss a smaller one.Looking through all the computed squared distances:From C1: 71, 38, 55, 40, 55, 14, 75From C2: 55, 46, 51, 64, 101, 32From C3: 101, 14, 47, 50, 39From C4: 91, 38, 71, 70From C5: 59, 34, 19From C6: 61, 48From C7: 73So, the smallest squared distances are 14 (twice), 19, 32, 34, 38, etc. So, 14 is indeed the smallest, occurring twice.Therefore, the answer is that both pairs (C1 & C7) and (C3 & C5) have the smallest Euclidean distance of sqrt(14). But since the question asks for "the pair", maybe it's expecting both pairs as the answer.Alternatively, perhaps I should present both as the solution.So, summarizing:1. Total unique pairings: 282. The pairs with the smallest Euclidean distance are C1 & C7 and C3 & C5, both with a distance of sqrt(14).But let me check if there's any pair with a smaller distance than 14. From the computations above, the next smallest is 19, so 14 is indeed the smallest.Therefore, the answer is that there are two pairs with the smallest distance: C1 & C7 and C3 & C5.However, the problem might expect a single pair, so perhaps I made a mistake in calculations. Let me double-check C3 & C5:C3: (4,5,6,1,3)C5: (2,4,7,3,5)Compute each difference:4-2=2, squared=45-4=1, squared=16-7=-1, squared=11-3=-2, squared=43-5=-2, squared=4Total: 4+1+1+4+4=14. Correct.Similarly, C1 & C7:C1: (3,7,2,5,4)C7: (1,6,4,7,3)Differences:3-1=2, squared=47-6=1, squared=12-4=-2, squared=45-7=-2, squared=44-3=1, squared=1Total: 4+1+4+4+1=14. Correct.So, both pairs are correct. Therefore, the answer is that both pairs have the smallest distance.But the problem might expect just one pair, so perhaps I need to see if there's a typo or if I misread the vectors.Wait, let me check the vectors again:C1: (3,7,2,5,4)C2: (6,1,3,2,8)C3: (4,5,6,1,3)C4: (7,2,1,8,6)C5: (2,4,7,3,5)C6: (8,3,5,6,2)C7: (1,6,4,7,3)C8: (5,2,8,4,7)Yes, those are correct. So, the calculations are correct.Therefore, the conclusion is that there are two pairs with the smallest Euclidean distance: C1 & C7 and C3 & C5.But the question says "the pair of cocktails", so maybe it's expecting both. Alternatively, perhaps I should present both as the answer.Alternatively, maybe I should present the pair with the smallest distance, and if there are multiple, list them all.So, in conclusion:1. The total number of unique pairings is 28.2. The pairs with the smallest Euclidean distance are C1 & C7 and C3 & C5, both with a distance of sqrt(14).But let me compute sqrt(14) to confirm it's approximately 3.7417.Yes, sqrt(14) ≈ 3.7417.Therefore, the final answers are:1. 28 unique pairings.2. The pairs C1 & C7 and C3 & C5 have the smallest Euclidean distance.But since the question asks for "the pair", perhaps it's expecting to list both pairs.Alternatively, maybe I should present the pair with the smallest distance, and if there are multiple, list them all.So, the answer is that both pairs (C1, C7) and (C3, C5) have the smallest Euclidean distance.But perhaps the problem expects just one pair, so maybe I made a mistake in calculations. Let me check again.Wait, perhaps I made a mistake in calculating the squared distances for C3 & C5.C3: (4,5,6,1,3)C5: (2,4,7,3,5)Differences:4-2=2, squared=45-4=1, squared=16-7=-1, squared=11-3=-2, squared=43-5=-2, squared=4Total: 4+1+1+4+4=14. Correct.Similarly, C1 & C7:C1: (3,7,2,5,4)C7: (1,6,4,7,3)Differences:3-1=2, squared=47-6=1, squared=12-4=-2, squared=45-7=-2, squared=44-3=1, squared=1Total: 4+1+4+4+1=14. Correct.So, both are correct. Therefore, the answer is that both pairs have the smallest distance.But the problem might expect a single pair, so perhaps I should check if there's a pair with a smaller distance than 14. From the computations above, the next smallest is 19, so 14 is indeed the smallest.Therefore, the answer is that both pairs (C1 & C7) and (C3 & C5) have the smallest Euclidean distance.But the question asks for "the pair of cocktails", so maybe it's expecting both pairs as the answer.Alternatively, perhaps the problem expects just one pair, but in reality, there are two pairs with the same minimal distance.Therefore, the answer is that the pairs C1 & C7 and C3 & C5 have the smallest Euclidean distance.But to be precise, the problem might expect just one pair, but since both have the same distance, both should be mentioned.So, in conclusion:1. The total number of unique pairings is 28.2. The pairs C1 & C7 and C3 & C5 have the smallest Euclidean distance of sqrt(14).Therefore, the final answers are:1. 282. C1 & C7 and C3 & C5But let me check if the problem expects a single pair or multiple. The problem says "the pair of cocktails", but in reality, there are two pairs. So, I think it's appropriate to mention both.Alternatively, perhaps I should present both pairs as the answer.Therefore, the final answer is:1. The total number of unique pairings is boxed{28}.2. The pairs with the smallest Euclidean distance are boxed{C_1 text{ and } C_7} and boxed{C_3 text{ and } C_5}.But since the problem might expect a single pair, perhaps I should check if there's a mistake. Alternatively, maybe I should present both pairs as the answer.Alternatively, perhaps I should present the pair with the smallest distance, and if there are multiple, list them all.Therefore, the answer is that both pairs (C1, C7) and (C3, C5) have the smallest Euclidean distance.But to be concise, I think the answer is that both pairs are the closest.Therefore, the final answer is:1. The total number of unique pairings is boxed{28}.2. The pairs with the smallest Euclidean distance are boxed{C_1 text{ and } C_7} and boxed{C_3 text{ and } C_5}.But perhaps the problem expects just one pair, so maybe I should present both as the answer.Alternatively, perhaps the problem expects just one pair, but in reality, there are two pairs with the same minimal distance.Therefore, the answer is that both pairs have the smallest distance.But to be precise, the problem might expect just one pair, but since both have the same distance, both should be mentioned.Therefore, the final answer is:1. The total number of unique pairings is boxed{28}.2. The pairs with the smallest Euclidean distance are boxed{C_1 text{ and } C_7} and boxed{C_3 text{ and } C_5}.</think>