🧠:<think>First, I need to determine the monthly savings from switching to e-books. The family currently spends 150 on physical books each month, and switching to e-readers reduces this expense by 70%. So, the monthly savings would be 70% of 150, which is 105.Next, I'll calculate the total initial cost of switching to e-readers. The e-reader costs 120, and each e-book averages 5. Assuming the family buys one e-book per month, the initial cost would be 120 plus 5, totaling 125.To find the break-even point, I'll divide the initial cost by the monthly savings: 125 divided by 105 per month, which equals approximately 1.19 months. Since the family can't break even in a fraction of a month, I'll round up to 2 months. This means it will take 2 months for the savings to offset the initial investment.For the second part, I need to calculate the future value of the savings after 5 years, starting from the month they break even. The monthly savings amount to 105, and the savings account offers an annual interest rate of 4%, compounded monthly. The monthly interest rate is 4% divided by 12, which is approximately 0.3333%.The number of months after breaking even is 5 years multiplied by 12, totaling 60 months. Using the future value formula for an ordinary annuity, FV = PMT × [(1 + r)^n - 1] / r, where PMT is 105, r is 0.3333%, and n is 60, the future value is calculated to be approximately 7,330.73.</think>

🧠:<think>Alright, so I've got this problem about modeling the growth rate of a moss species using a differential equation. It's split into two parts, and I need to figure out the equilibrium points and then the integral of the growth rate over a period. Let me start with the first part.Problem 1: Finding Equilibrium PointsThe differential equation given is:[frac{dG}{dt} = alpha G(t) lnleft(frac{L(t) + k}{H(t) + k}right) - beta G(t)^2]Equilibrium points occur where the growth rate ( frac{dG}{dt} = 0 ). So, I need to set the right-hand side of the equation to zero and solve for ( G(t) ).Setting ( frac{dG}{dt} = 0 ):[alpha G lnleft(frac{L + k}{H + k}right) - beta G^2 = 0]Let me factor out ( G ):[G left( alpha lnleft(frac{L + k}{H + k}right) - beta G right) = 0]So, the solutions are when either ( G = 0 ) or the term in the parentheses is zero.Case 1: ( G = 0 )This is one equilibrium point. It represents a state where the moss isn't growing at all, perhaps because the environmental conditions are too harsh or the moss isn't present.Case 2: ( alpha lnleft(frac{L + k}{H + k}right) - beta G = 0 )Solving for ( G ):[beta G = alpha lnleft(frac{L + k}{H + k}right)][G = frac{alpha}{beta} lnleft(frac{L + k}{H + k}right)]So, the non-zero equilibrium point is ( G = frac{alpha}{beta} lnleft(frac{L + k}{H + k}right) ).But wait, ( L(t) ) and ( H(t) ) are functions of time. So, are these equilibrium points time-dependent? Hmm, in the context of equilibrium points for a differential equation, they are typically steady states where the system doesn't change with time. However, in this case, since ( L(t) ) and ( H(t) ) are time-dependent, the equilibrium points are also functions of time.But the question says "derive the expression for the equilibrium points of the growth rate ( G(t) ) in terms of the given parameters." So, perhaps they just want the expression in terms of ( L(t) ) and ( H(t) ), recognizing that these are functions of time.So, the equilibrium points are:1. ( G = 0 )2. ( G = frac{alpha}{beta} lnleft(frac{L(t) + k}{H(t) + k}right) )But let me think again. If ( L(t) ) and ( H(t) ) are given as functions, then the equilibrium points are also functions of time. So, in a sense, for each time ( t ), there is an equilibrium ( G(t) ). But in dynamical systems, equilibrium points are typically constant solutions. So, if ( L(t) ) and ( H(t) ) are not constants, then the system doesn't have constant equilibrium points, but rather time-dependent ones.But the problem might be asking for the form of the equilibrium solutions, regardless of whether they're constant or not. So, perhaps the answer is as above.Alternatively, if we consider that the system might reach a steady state where ( G(t) ) is periodic, matching the periodicity of ( L(t) ) and ( H(t) ), but that might be more related to part 2.Wait, part 2 talks about the integral over a period when ( G(t) ) reaches a periodic steady state. So, maybe in part 1, they just want the expression for equilibrium points, which are functions of ( L(t) ) and ( H(t) ).So, to recap, equilibrium points are:1. ( G = 0 )2. ( G = frac{alpha}{beta} lnleft( frac{L(t) + k}{H(t) + k} right) )But let me check if ( G ) can be negative. Since ( ln ) can be negative if ( frac{L + k}{H + k} < 1 ), so ( G ) could be negative, but growth rate can't be negative. Hmm, that's a problem. So, perhaps the model assumes that ( G(t) ) is non-negative, so the logarithm must be non-negative as well.Therefore, ( frac{L(t) + k}{H(t) + k} geq 1 ), which implies ( L(t) geq H(t) ). Otherwise, the logarithm is negative, leading to a negative growth rate, which doesn't make sense biologically.So, perhaps the model is only valid when ( L(t) geq H(t) ), ensuring that ( G(t) ) is non-negative.Alternatively, the model might allow for negative growth rates, meaning the moss could be declining, but in the context of growth rate, it's usually considered as a positive quantity. So, maybe the equilibrium points are only valid when ( lnleft( frac{L + k}{H + k} right) geq 0 ), i.e., ( L geq H ).But the problem statement doesn't specify that, so perhaps we just proceed with the mathematical solution.So, the equilibrium points are ( G = 0 ) and ( G = frac{alpha}{beta} lnleft( frac{L(t) + k}{H(t) + k} right) ).But since ( L(t) ) and ( H(t) ) are given as functions, perhaps we need to express the equilibrium points in terms of the given parameters, substituting ( L(t) = L_0 e^{-lambda t} ) and ( H(t) = H_0 + mu sin(omega t) ).So, substituting these into the expression:[G = frac{alpha}{beta} lnleft( frac{L_0 e^{-lambda t} + k}{H_0 + mu sin(omega t) + k} right)]So, that's the expression for the non-zero equilibrium point.But wait, equilibrium points are typically constant solutions, but here, since ( L(t) ) and ( H(t) ) are time-dependent, the equilibrium points are also time-dependent. So, perhaps in this context, the equilibrium points are not constant but vary with time as per the environmental factors.Therefore, the equilibrium points are:1. ( G(t) = 0 )2. ( G(t) = frac{alpha}{beta} lnleft( frac{L_0 e^{-lambda t} + k}{H_0 + mu sin(omega t) + k} right) )So, that's the expression for the equilibrium points in terms of the given parameters.Problem 2: Integral of Growth Rate over Period [0, T]The integral ( I ) is given by:[I = int_0^T G(t) , dt]Assuming ( G(t) ) reaches a periodic steady state, we need to express ( I ) in terms of the average values of ( L(t) ) and ( H(t) ) over the interval [0, T].First, let's recall that if a function ( f(t) ) is periodic with period ( T ), then the average value over one period is:[langle f rangle = frac{1}{T} int_0^T f(t) , dt]So, the average light intensity ( langle L rangle ) and average humidity ( langle H rangle ) can be defined as:[langle L rangle = frac{1}{T} int_0^T L(t) , dt = frac{1}{T} int_0^T L_0 e^{-lambda t} , dt][langle H rangle = frac{1}{T} int_0^T H(t) , dt = frac{1}{T} int_0^T (H_0 + mu sin(omega t)) , dt]Calculating these averages:For ( langle L rangle ):[langle L rangle = frac{L_0}{T} int_0^T e^{-lambda t} , dt = frac{L_0}{T} left[ frac{e^{-lambda t}}{-lambda} right]_0^T = frac{L_0}{T} left( frac{1 - e^{-lambda T}}{lambda} right)]For ( langle H rangle ):[langle H rangle = frac{1}{T} int_0^T H_0 , dt + frac{mu}{T} int_0^T sin(omega t) , dt = H_0 + frac{mu}{T} left[ frac{-cos(omega t)}{omega} right]_0^T][= H_0 + frac{mu}{T omega} left( -cos(omega T) + cos(0) right) = H_0 + frac{mu}{T omega} (1 - cos(omega T))]Now, assuming ( G(t) ) reaches a periodic steady state, meaning that ( G(t) ) is periodic with the same period ( T ) as ( L(t) ) and ( H(t) ). Therefore, the average growth rate ( langle G rangle ) is:[langle G rangle = frac{1}{T} int_0^T G(t) , dt]But the integral ( I ) is:[I = int_0^T G(t) , dt = T langle G rangle]So, if we can express ( langle G rangle ) in terms of ( langle L rangle ) and ( langle H rangle ), then ( I ) can be expressed as ( T langle G rangle ).However, the differential equation is nonlinear, so the average of ( G(t) ) isn't directly the same as plugging the averages into the equation. But perhaps under certain conditions, like if the system is linear, we could do that, but since it's nonlinear, it's more complicated.Wait, but the problem says "assuming ( G(t) ) reaches a periodic steady state." So, in a periodic steady state, the time average of ( G(t) ) can be related to the time averages of ( L(t) ) and ( H(t) ).But how?Let me think. If ( G(t) ) is in a periodic steady state, then over each period, the integral of ( dG/dt ) is zero because it returns to the same value. So, integrating the differential equation over one period:[int_0^T frac{dG}{dt} , dt = 0][int_0^T left[ alpha G(t) lnleft(frac{L(t) + k}{H(t) + k}right) - beta G(t)^2 right] dt = 0]So,[alpha int_0^T G(t) lnleft(frac{L(t) + k}{H(t) + k}right) dt - beta int_0^T G(t)^2 dt = 0]But this seems complicated because it involves the integral of ( G(t) ) times the logarithm term, and the integral of ( G(t)^2 ).Alternatively, perhaps we can use the concept of averaging. If ( G(t) ) is periodic, then we can consider the average of the differential equation over one period.Taking the average of both sides:[langle frac{dG}{dt} rangle = alpha langle G lnleft(frac{L + k}{H + k}right) rangle - beta langle G^2 rangle = 0]So,[alpha langle G lnleft(frac{L + k}{H + k}right) rangle = beta langle G^2 rangle]But this still involves the average of ( G ln(cdot) ), which isn't straightforward to express in terms of ( langle G rangle ), ( langle L rangle ), and ( langle H rangle ).Alternatively, if we assume that ( G(t) ) is approximately constant over the period, or that the variations are small, we might approximate ( G(t) approx langle G rangle ), and similarly for the logarithm term. But this is a rough approximation.Wait, but the problem says "express ( I ) in terms of the average values of the functions ( L(t) ) and ( H(t) ) over the interval [0, T]." So, perhaps we can express ( I ) as ( T times ) average growth rate, and relate the average growth rate to the average ( L ) and ( H ).But without knowing more about the form of ( G(t) ), it's tricky. Maybe we can make an approximation.Suppose that ( G(t) ) is such that in the steady state, the time average of ( G(t) ) satisfies:[langle G rangle = frac{alpha}{beta} langle lnleft( frac{L + k}{H + k} right) rangle]But that's assuming that ( G(t) ) is proportional to the logarithm term, which might not hold because of the nonlinearity.Alternatively, perhaps in the steady state, the growth rate ( G(t) ) is such that the average of the right-hand side of the differential equation is zero, but that's not directly helpful.Wait, going back to the integral over the period:[int_0^T frac{dG}{dt} dt = 0][int_0^T left[ alpha G(t) lnleft( frac{L(t) + k}{H(t) + k} right) - beta G(t)^2 right] dt = 0]So,[alpha int_0^T G(t) lnleft( frac{L(t) + k}{H(t) + k} right) dt = beta int_0^T G(t)^2 dt]But this doesn't directly give us ( I = int G(t) dt ). However, if we can relate ( int G(t) dt ) to the averages of ( L ) and ( H ), perhaps through some manipulation.Alternatively, perhaps we can consider that in the steady state, the growth rate ( G(t) ) is such that the integral of ( G(t) ) over the period is proportional to the integral of the logarithm term.But I'm not sure. Maybe another approach is needed.Wait, the problem says "express ( I ) in terms of the average values of the functions ( L(t) ) and ( H(t) ) over the interval [0, T]." So, perhaps we can write ( I ) as ( T times ) average ( G(t) ), and then express average ( G(t) ) in terms of average ( L(t) ) and average ( H(t) ).But how?Alternatively, perhaps we can use the fact that in the steady state, the time average of ( G(t) ) satisfies a certain equation. Let me think.If ( G(t) ) is periodic with period ( T ), then over the period, the integral of ( dG/dt ) is zero, which we've already used.But perhaps we can also consider the average of the differential equation:[langle frac{dG}{dt} rangle = alpha langle G lnleft( frac{L + k}{H + k} right) rangle - beta langle G^2 rangle = 0]So,[alpha langle G lnleft( frac{L + k}{H + k} right) rangle = beta langle G^2 rangle]But without knowing more about the correlation between ( G ) and the logarithm term, it's hard to proceed.Alternatively, perhaps we can make a linear approximation, assuming that ( G(t) ) is small, so that ( G(t)^2 ) is negligible, but that might not be the case.Alternatively, if ( G(t) ) is in a steady state, perhaps the integral ( I = int_0^T G(t) dt ) can be expressed as ( frac{alpha}{beta} int_0^T lnleft( frac{L(t) + k}{H(t) + k} right) dt ), but that would be if ( G(t) ) were equal to ( frac{alpha}{beta} ln(...) ), which is the non-zero equilibrium point. But since ( G(t) ) is in a periodic steady state, it's not necessarily equal to that equilibrium point at all times.Wait, but in a steady state, perhaps the time average of ( G(t) ) is equal to the average of the equilibrium expression. That is:[langle G rangle = frac{alpha}{beta} langle lnleft( frac{L + k}{H + k} right) rangle]But that's an assumption. If that's the case, then:[I = T langle G rangle = frac{alpha}{beta} T langle lnleft( frac{L + k}{H + k} right) rangle]But the problem asks to express ( I ) in terms of the average values of ( L(t) ) and ( H(t) ), not the average of the logarithm. So, perhaps we need to relate ( langle ln(...) rangle ) to ( langle L rangle ) and ( langle H rangle ).But that's not straightforward because the logarithm of a ratio isn't the same as the ratio of logarithms or the difference of logarithms. So, unless we can linearize the logarithm term, which might not be accurate.Alternatively, perhaps we can use a Taylor expansion or some approximation for the logarithm.Let me consider that ( lnleft( frac{L + k}{H + k} right) = ln(L + k) - ln(H + k) ). So,[langle lnleft( frac{L + k}{H + k} right) rangle = langle ln(L + k) rangle - langle ln(H + k) rangle]But unless we can relate ( langle ln(L + k) rangle ) to ( langle L rangle ), which isn't straightforward, this might not help.Alternatively, if ( k ) is a small positive constant, as given, perhaps ( L(t) ) and ( H(t) ) are much larger than ( k ), so we can approximate ( ln(L + k) approx ln L + frac{k}{L} ), using the expansion ( ln(a + b) approx ln a + frac{b}{a} ) for small ( b ).So,[ln(L + k) approx ln L + frac{k}{L}][ln(H + k) approx ln H + frac{k}{H}]Therefore,[lnleft( frac{L + k}{H + k} right) approx lnleft( frac{L}{H} right) + frac{k}{L} - frac{k}{H}]So, the average becomes:[langle lnleft( frac{L + k}{H + k} right) rangle approx langle lnleft( frac{L}{H} right) rangle + k left( langle frac{1}{L} rangle - langle frac{1}{H} rangle right)]But again, unless we can relate ( langle ln(L/H) rangle ) to ( langle L rangle ) and ( langle H rangle ), which isn't straightforward, this might not help.Alternatively, perhaps the problem expects a simpler approach, assuming that the logarithm can be expressed in terms of the averages. But that's not mathematically rigorous.Wait, perhaps the problem is suggesting that in the steady state, the integral ( I ) can be approximated by integrating the equilibrium expression over time. That is, if ( G(t) ) is close to its equilibrium value, then:[I approx int_0^T frac{alpha}{beta} lnleft( frac{L(t) + k}{H(t) + k} right) dt]But that's assuming ( G(t) ) is always at equilibrium, which might not be the case, but perhaps in a steady state, it's fluctuating around the equilibrium.Alternatively, perhaps the integral ( I ) can be expressed as:[I = frac{alpha}{beta} int_0^T lnleft( frac{L(t) + k}{H(t) + k} right) dt]But then, how does this relate to the average values of ( L(t) ) and ( H(t) )?Wait, maybe the problem is expecting us to recognize that if ( G(t) ) is in a periodic steady state, then the integral ( I ) is equal to the integral of the equilibrium expression over the period, which would be:[I = int_0^T frac{alpha}{beta} lnleft( frac{L(t) + k}{H(t) + k} right) dt]But then, to express this in terms of the average values, we can write:[I = frac{alpha}{beta} T cdot langle lnleft( frac{L + k}{H + k} right) rangle]But the problem says "express ( I ) in terms of the average values of the functions ( L(t) ) and ( H(t) )", not the average of the logarithm. So, perhaps we need to find a way to express ( langle ln(...) rangle ) in terms of ( langle L rangle ) and ( langle H rangle ).But unless we make an approximation or assume something about the distribution of ( L(t) ) and ( H(t) ), this isn't possible. Maybe the problem expects us to recognize that the integral ( I ) is proportional to the integral of the logarithm term, which can be expressed as the average of the logarithm times ( T ).But since the problem specifically mentions expressing ( I ) in terms of the average values of ( L(t) ) and ( H(t) ), perhaps we need to consider that the logarithm can be approximated by its average.Wait, perhaps if we assume that ( lnleft( frac{L + k}{H + k} right) ) can be approximated by ( lnleft( frac{langle L rangle + k}{langle H rangle + k} right) ), then:[I approx frac{alpha}{beta} T lnleft( frac{langle L rangle + k}{langle H rangle + k} right)]But this is an approximation because the average of a function isn't necessarily the function of the average, unless the function is linear, which the logarithm isn't.However, given that the problem asks to express ( I ) in terms of the average values, perhaps this is the intended approach, even though it's an approximation.So, putting it all together, the integral ( I ) can be expressed as:[I = frac{alpha}{beta} T lnleft( frac{langle L rangle + k}{langle H rangle + k} right)]where ( langle L rangle ) and ( langle H rangle ) are the average values of ( L(t) ) and ( H(t) ) over [0, T].Now, discussing the conditions under which ( I ) is maximized. Since ( I ) is proportional to the logarithm term, to maximize ( I ), we need to maximize ( lnleft( frac{langle L rangle + k}{langle H rangle + k} right) ), which is equivalent to maximizing ( frac{langle L rangle + k}{langle H rangle + k} ).Therefore, ( I ) is maximized when ( langle L rangle ) is as large as possible and ( langle H rangle ) is as small as possible.Looking at the expressions for ( langle L rangle ) and ( langle H rangle ):- ( langle L rangle = frac{L_0}{lambda T} (1 - e^{-lambda T}) ). To maximize ( langle L rangle ), we need to maximize ( L_0 ) and minimize ( lambda ) (since a smaller ( lambda ) means slower decay of light intensity over time).- ( langle H rangle = H_0 + frac{mu}{T omega} (1 - cos(omega T)) ). To minimize ( langle H rangle ), we need to minimize ( H_0 ) and ( mu ), and perhaps choose ( omega ) such that ( cos(omega T) ) is maximized, which would be when ( omega T = 2pi n ) for integer ( n ), making ( cos(omega T) = 1 ), thus minimizing the second term.Therefore, ( I ) is maximized when:- Light intensity ( L(t) ) is as high as possible on average, which occurs when ( L_0 ) is large and ( lambda ) is small (slow decay).- Humidity ( H(t) ) is as low as possible on average, which occurs when ( H_0 ) is small, ( mu ) is small (less variability), and the frequency ( omega ) is such that the oscillation completes an integer number of cycles over the period ( T ), maximizing ( cos(omega T) ) and thus minimizing the added variability.So, in summary, ( I ) is maximized when the average light intensity is maximized and the average humidity is minimized, considering the parameters that control their respective functions.Final Answer1. The equilibrium points are ( boxed{0} ) and ( boxed{frac{alpha}{beta} lnleft( frac{L_0 e^{-lambda t} + k}{H_0 + mu sin(omega t) + k} right)} ).2. The integral ( I ) is expressed as ( boxed{frac{alpha}{beta} T lnleft( frac{langle L rangle + k}{langle H rangle + k} right)} ), where ( langle L rangle ) and ( langle H rangle ) are the average light intensity and humidity over the interval. ( I ) is maximized when ( langle L rangle ) is maximized and ( langle H rangle ) is minimized.</think>

🧠:<think>Alright, so I have this problem where Samantha gains comfort from watching a movie and then discussing it with her social worker, Alex. There are two parts to this problem: calculating the comfort from watching the movie and then from discussing it, and then adding both together for the total comfort. Let me try to tackle each part step by step.Starting with the first part: calculating the total comfort Samantha gains after watching a movie for 2 hours. The comfort level is modeled by a piecewise function, which means the function has different expressions depending on the time interval. The function is given as:[ C(t) =begin{cases} 3t^2 + 2t + 1 & text{if } 0 leq t leq 1, -2t^3 + 3t^2 + t + 2 & text{if } 1 < t leq 2end{cases}]So, since the movie is watched for 2 hours, we need to calculate the comfort from both intervals: from 0 to 1 hour and from 1 to 2 hours. I think this means we have to compute the integral of C(t) from 0 to 2, but since it's piecewise, we'll split it into two integrals: from 0 to 1 and from 1 to 2.First, let me recall how to integrate a function. The integral of a function over an interval gives the area under the curve, which in this case represents the total comfort gained over that time period.Starting with the first interval, 0 to 1 hour, the function is ( 3t^2 + 2t + 1 ). Let me write that down:Integral from 0 to 1 of ( 3t^2 + 2t + 1 ) dt.To integrate term by term:- Integral of ( 3t^2 ) is ( t^3 ) because ( int t^n dt = frac{t^{n+1}}{n+1} ), so ( 3 times frac{t^{3}}{3} = t^3 ).- Integral of ( 2t ) is ( t^2 ) because ( 2 times frac{t^{2}}{2} = t^2 ).- Integral of 1 is ( t ) because ( int 1 dt = t ).So putting it all together, the integral becomes:[ left[ t^3 + t^2 + t right] ] evaluated from 0 to 1.Calculating at t=1:( 1^3 + 1^2 + 1 = 1 + 1 + 1 = 3 ).Calculating at t=0:( 0^3 + 0^2 + 0 = 0 ).So the integral from 0 to 1 is ( 3 - 0 = 3 ).Okay, that seems straightforward. Now moving on to the second interval, from 1 to 2 hours, the function is ( -2t^3 + 3t^2 + t + 2 ).So we need to compute the integral from 1 to 2 of ( -2t^3 + 3t^2 + t + 2 ) dt.Again, integrating term by term:- Integral of ( -2t^3 ) is ( -frac{2}{4}t^4 = -frac{1}{2}t^4 ).- Integral of ( 3t^2 ) is ( t^3 ) because ( 3 times frac{t^3}{3} = t^3 ).- Integral of ( t ) is ( frac{1}{2}t^2 ).- Integral of 2 is ( 2t ).Putting it all together, the integral becomes:[ left[ -frac{1}{2}t^4 + t^3 + frac{1}{2}t^2 + 2t right] ] evaluated from 1 to 2.Let me compute this at t=2 first:- ( -frac{1}{2}(2)^4 = -frac{1}{2}(16) = -8 )- ( (2)^3 = 8 )- ( frac{1}{2}(2)^2 = frac{1}{2}(4) = 2 )- ( 2(2) = 4 )Adding these together: ( -8 + 8 + 2 + 4 = 6 ).Now at t=1:- ( -frac{1}{2}(1)^4 = -frac{1}{2}(1) = -0.5 )- ( (1)^3 = 1 )- ( frac{1}{2}(1)^2 = frac{1}{2}(1) = 0.5 )- ( 2(1) = 2 )Adding these together: ( -0.5 + 1 + 0.5 + 2 = 3 ).So the integral from 1 to 2 is ( 6 - 3 = 3 ).Therefore, the total comfort from watching the movie for 2 hours is the sum of the two integrals: 3 (from 0 to 1) + 3 (from 1 to 2) = 6.Wait, hold on. That seems a bit too clean. Let me double-check my calculations.For the first integral from 0 to 1:- Integral of ( 3t^2 ) is indeed ( t^3 ).- Integral of ( 2t ) is ( t^2 ).- Integral of 1 is ( t ).Evaluated at 1: 1 + 1 + 1 = 3. At 0: 0. So 3 - 0 = 3. That seems correct.For the second integral from 1 to 2:- Integral of ( -2t^3 ) is ( -frac{1}{2}t^4 ).- Integral of ( 3t^2 ) is ( t^3 ).- Integral of ( t ) is ( frac{1}{2}t^2 ).- Integral of 2 is ( 2t ).Evaluated at 2: ( -frac{1}{2}(16) + 8 + 2 + 4 = -8 + 8 + 2 + 4 = 6 ).Evaluated at 1: ( -frac{1}{2}(1) + 1 + 0.5 + 2 = -0.5 + 1 + 0.5 + 2 = 3 ).So 6 - 3 = 3. That also seems correct.Adding both parts: 3 + 3 = 6. So the total comfort from watching the movie is 6. Hmm, okay.Moving on to the second part: determining the total comfort gained from 1 hour of discussion. The function given is ( D(t) = 5e^{-0.5t} ), where t is the time in hours. So we need to compute the integral of D(t) from 0 to 1.Integral from 0 to 1 of ( 5e^{-0.5t} ) dt.I remember that the integral of ( e^{kt} ) is ( frac{1}{k}e^{kt} ). So here, k is -0.5, so the integral should be ( frac{5}{-0.5}e^{-0.5t} ) evaluated from 0 to 1.Wait, let me write that out:Integral of ( 5e^{-0.5t} ) dt is ( 5 times frac{1}{-0.5} e^{-0.5t} + C ), which simplifies to ( -10e^{-0.5t} + C ).So evaluating from 0 to 1:At t=1: ( -10e^{-0.5(1)} = -10e^{-0.5} ).At t=0: ( -10e^{-0.5(0)} = -10e^{0} = -10(1) = -10 ).So the integral is ( (-10e^{-0.5}) - (-10) = -10e^{-0.5} + 10 = 10(1 - e^{-0.5}) ).Now, let me compute the numerical value of this. Since ( e^{-0.5} ) is approximately ( e^{-0.5} approx 0.6065 ).So ( 1 - 0.6065 = 0.3935 ).Multiplying by 10: ( 10 times 0.3935 = 3.935 ).So approximately 3.935 units of comfort from the discussion.Wait, but let me make sure I didn't make a mistake in the integral.The integral of ( e^{kt} ) is ( frac{1}{k}e^{kt} ). So for ( e^{-0.5t} ), the integral is ( frac{1}{-0.5}e^{-0.5t} = -2e^{-0.5t} ). Then multiplied by 5, it's ( -10e^{-0.5t} ). So that's correct.Evaluating from 0 to 1:At 1: ( -10e^{-0.5} ).At 0: ( -10e^{0} = -10 ).So the definite integral is ( (-10e^{-0.5}) - (-10) = -10e^{-0.5} + 10 = 10(1 - e^{-0.5}) ). That's correct.Calculating ( e^{-0.5} ) is approximately 0.6065, so 1 - 0.6065 is 0.3935, times 10 is 3.935. So approximately 3.935.But since the question says to determine the total comfort, maybe we can leave it in exact terms? Or do they want a numerical value? The problem statement doesn't specify, but since the first part was exact, maybe we should do the same here.So ( 10(1 - e^{-0.5}) ) is the exact value. Alternatively, if we need a decimal, it's approximately 3.935.But let me check if the question says anything about the form. It says "determine the total comfort gained from 1 hour of discussion." It doesn't specify, so perhaps we can present both, but in the final answer, maybe they want a numerical value.But let's see, in the first part, the answer was an integer, 6. Maybe they expect an exact value here as well, so 10(1 - e^{-0.5}) is exact.Alternatively, maybe they want it in terms of e, but 10(1 - e^{-0.5}) is as simplified as it gets.So, moving on, the total comfort is the sum of the comfort from watching the movie and discussing it.From watching: 6.From discussing: 10(1 - e^{-0.5}) or approximately 3.935.So total comfort is 6 + 10(1 - e^{-0.5}) or approximately 6 + 3.935 = 9.935.But again, the question says to combine the results from the two sub-problems. It doesn't specify whether to present it as an exact value or approximate. Since the first part was exact, and the second part can be exact, perhaps we should present it as 6 + 10(1 - e^{-0.5}).But let me compute 10(1 - e^{-0.5}) exactly:10(1 - e^{-0.5}) = 10 - 10e^{-0.5}.So total comfort is 6 + 10 - 10e^{-0.5} = 16 - 10e^{-0.5}.Alternatively, if we factor 10, it's 16 - 10e^{-0.5}.But let me see if that's the case.Wait, no, 6 + 10(1 - e^{-0.5}) is 6 + 10 - 10e^{-0.5} = 16 - 10e^{-0.5}. Yes, that's correct.Alternatively, if we leave it as 6 + 10(1 - e^{-0.5}), that's also acceptable.But perhaps 16 - 10e^{-0.5} is a more simplified exact form.Alternatively, if they want a numerical value, 16 - 10*(0.6065) = 16 - 6.065 = 9.935, which is approximately 9.935.But let me check if the problem expects an exact answer or a decimal. The first part was an exact integer, so maybe the second part is also expected to be exact, so 16 - 10e^{-0.5}.Alternatively, since the first part was 6, which is exact, and the second part is 10(1 - e^{-0.5}), which is also exact, adding them together gives 6 + 10(1 - e^{-0.5}) = 16 - 10e^{-0.5}.Yes, that seems correct.Wait, but let me double-check the integral for the discussion part again because sometimes constants can be tricky.Given D(t) = 5e^{-0.5t}, integral from 0 to 1.Integral is ( int_{0}^{1} 5e^{-0.5t} dt ).Let me make substitution: let u = -0.5t, then du/dt = -0.5, so dt = du / (-0.5) = -2du.So integral becomes:5 * ∫ e^{u} * (-2) du from u = 0 to u = -0.5.Wait, when t=0, u=0; when t=1, u=-0.5.So the integral becomes:5 * (-2) ∫_{0}^{-0.5} e^{u} du = -10 [e^{u}] from 0 to -0.5 = -10 (e^{-0.5} - e^{0}) = -10 (e^{-0.5} - 1) = -10e^{-0.5} + 10 = 10(1 - e^{-0.5}).Yes, that's the same result as before. So that's correct.Therefore, the total comfort from discussion is 10(1 - e^{-0.5}), and total comfort overall is 6 + 10(1 - e^{-0.5}) = 16 - 10e^{-0.5}.Alternatively, if we want to write it as 16 - 10e^{-1/2}, since 0.5 is 1/2.But in any case, that's the exact value.Alternatively, if we compute it numerically, as I did earlier, it's approximately 9.935.But since the problem didn't specify, I think it's safer to present the exact value, which is 16 - 10e^{-0.5}.Alternatively, if they expect a decimal, maybe 9.935.But let me check the problem statement again."Finally, combine the results from the two sub-problems to find the total comfort Samantha gains from both watching the movie for 2 hours and discussing it for 1 hour."It doesn't specify the form, but since the first part was an exact integer and the second part is an exact expression, perhaps the total should be presented as 16 - 10e^{-0.5}.Alternatively, if they expect a numerical value, maybe 9.935.But to be thorough, let me compute 16 - 10e^{-0.5} numerically.Compute e^{-0.5} ≈ 0.60653066.So 10 * 0.60653066 ≈ 6.0653066.So 16 - 6.0653066 ≈ 9.9346934.So approximately 9.935.So, depending on what's expected, either 16 - 10e^{-0.5} or approximately 9.935.But let me check if the problem expects the answer in a specific form. The first part was an exact integral, resulting in 6, which is an integer. The second part is an integral resulting in 10(1 - e^{-0.5}), which is exact. So adding them together, 6 + 10(1 - e^{-0.5}) = 16 - 10e^{-0.5}.Alternatively, if they want a decimal, 9.935.But since the problem didn't specify, perhaps we can present both, but in the final answer, maybe they expect the exact form.Alternatively, maybe I should present both results as exact and then add them as exact, so 6 + 10(1 - e^{-0.5}) = 16 - 10e^{-0.5}.Alternatively, if they expect a numerical value, 9.935.But to be safe, perhaps I should present both, but since the problem didn't specify, maybe the exact form is better.Wait, but in the first part, the integral was 3 + 3 = 6, which is exact. The second part is 10(1 - e^{-0.5}), which is exact. So adding them together, 6 + 10(1 - e^{-0.5}) = 16 - 10e^{-0.5}.Alternatively, if they want a numerical value, 9.935.But let me check if the problem expects an exact answer or a decimal. Since the first part was an exact integer, and the second part is an exact expression, perhaps the total should be presented as 16 - 10e^{-0.5}.Alternatively, if they expect a decimal, maybe 9.935.But to be thorough, let me compute 16 - 10e^{-0.5} numerically.Compute e^{-0.5} ≈ 0.60653066.So 10 * 0.60653066 ≈ 6.0653066.So 16 - 6.0653066 ≈ 9.9346934.So approximately 9.935.Therefore, the total comfort is approximately 9.935.But since the problem didn't specify, I think it's better to present the exact value, which is 16 - 10e^{-0.5}.Alternatively, if they expect a decimal, 9.935.But to be safe, I'll present both.Wait, but in the problem statement, the first part was an exact integral, so 6, and the second part is an exact integral, so 10(1 - e^{-0.5}), so adding them together, 6 + 10(1 - e^{-0.5}) = 16 - 10e^{-0.5}.Alternatively, if they want a numerical value, 9.935.But let me check if the problem expects the answer in a specific form. The problem says "combine the results from the two sub-problems to find the total comfort." It doesn't specify, so perhaps we can present the exact form.Alternatively, maybe they expect a decimal. Let me see if the problem mentions anything about decimal places or exact values. It doesn't, so perhaps both are acceptable.But in the first part, the answer was an integer, so maybe they expect an exact value here as well, so 16 - 10e^{-0.5}.Alternatively, if they want a decimal, 9.935.But to be thorough, I'll compute it as 16 - 10e^{-0.5} ≈ 9.935.So, in conclusion, the total comfort Samantha gains is approximately 9.935 units.But let me make sure I didn't make any mistakes in the integrals.First integral from 0 to 1: 3t² + 2t + 1. Integral is t³ + t² + t. Evaluated at 1: 1 + 1 + 1 = 3. At 0: 0. So 3. Correct.Second integral from 1 to 2: -2t³ + 3t² + t + 2. Integral is -0.5t⁴ + t³ + 0.5t² + 2t. Evaluated at 2: -0.5*(16) + 8 + 0.5*(4) + 4 = -8 + 8 + 2 + 4 = 6. Evaluated at 1: -0.5*(1) + 1 + 0.5*(1) + 2 = -0.5 + 1 + 0.5 + 2 = 3. So 6 - 3 = 3. Correct.Total from watching: 3 + 3 = 6. Correct.Discussion integral: 5e^{-0.5t} from 0 to 1. Integral is -10e^{-0.5t} evaluated from 0 to 1: -10e^{-0.5} - (-10) = 10(1 - e^{-0.5}). Correct.So total comfort: 6 + 10(1 - e^{-0.5}) = 16 - 10e^{-0.5} ≈ 9.935.Yes, that seems correct.So, to summarize:1. Comfort from watching: 6.2. Comfort from discussing: 10(1 - e^{-0.5}) ≈ 3.935.Total comfort: 6 + 3.935 ≈ 9.935.Alternatively, exact form: 16 - 10e^{-0.5}.I think that's it.</think>

🧠:<think>Okay, so I'm trying to solve this problem about a geological formation that's shaped like a truncated cone, also known as a frustum. The formation has a height of 100 meters, a top radius of 30 meters, and a bottom radius of 50 meters. We took a rock sample from a cross-section at a height of 40 meters from the base, and I need to figure out two things: the radius of that cross-section and the area of that cross-section. Then, I also need to calculate the volume of the part of the formation above that cross-section.Alright, let's start with the first part: finding the radius at 40 meters from the base. Since it's a truncated cone, it's like a cone that's been cut off from the top. So, I think I can model this as a larger cone minus a smaller cone. The original larger cone would have a height of 100 meters plus the height of the smaller cone that was removed. Hmm, but wait, actually, the truncated cone itself is 100 meters tall, with the top radius being 30 meters and the bottom radius being 50 meters. So, maybe I can think of it as part of a larger cone.Let me visualize this. If we imagine the truncated cone as part of a larger cone, then the height of the larger cone would be the height of the truncated cone plus the height of the smaller cone that was cut off. Let me denote the height of the larger cone as H, and the height of the smaller cone as h. So, H = 100 + h. The radius of the larger cone would be 50 meters, and the radius of the smaller cone would be 30 meters. Since these are similar cones, their dimensions are proportional.So, the ratio of the radii should be equal to the ratio of their heights. That is, 30/50 = h/(100 + h). Let me write that down:30/50 = h/(100 + h)Simplifying 30/50, that's 3/5. So,3/5 = h/(100 + h)Cross-multiplying, we get:3*(100 + h) = 5*h300 + 3h = 5hSubtracting 3h from both sides:300 = 2hSo, h = 150 meters.Wait, that means the height of the smaller cone is 150 meters, and the height of the larger cone is H = 100 + 150 = 250 meters. Hmm, that seems a bit counterintuitive because the truncated cone is only 100 meters tall, but the smaller cone is 150 meters. Maybe I made a mistake in setting up the ratio.Let me think again. The truncated cone is between the heights of h and H, where H is the total height of the original cone. So, the radius at the base of the truncated cone is 50 meters, which corresponds to the height H, and the radius at the top of the truncated cone is 30 meters, which corresponds to the height H - 100 meters. So, the ratio of radii is 30/50 = (H - 100)/H.So, 30/50 = (H - 100)/HSimplify 30/50 to 3/5:3/5 = (H - 100)/HCross-multiplying:3H = 5(H - 100)3H = 5H - 500Subtract 3H from both sides:0 = 2H - 500So, 2H = 500H = 250 meters.Okay, so that's consistent with what I had before. So, the original cone is 250 meters tall, and the truncated cone is the part from 150 meters to 250 meters. So, the radius at any height y from the base of the original cone would be (y/H)*R, where R is 50 meters. So, at a height of 40 meters from the base, which is within the truncated cone, we need to find the radius.Wait, no. The height of 40 meters is from the base of the truncated cone, which is at 150 meters from the apex of the original cone. So, if we take a cross-section at 40 meters from the base of the truncated cone, that would be at a height of 150 + 40 = 190 meters from the apex of the original cone.So, the radius at that height would be (190/H)*R = (190/250)*50.Calculating that:190/250 = 0.760.76 * 50 = 38 meters.Wait, so the radius at 40 meters from the base of the truncated cone is 38 meters. That seems reasonable because it's between 30 and 50 meters.Let me verify this another way. Since the truncated cone has a height of 100 meters, and the radius changes from 50 meters at the base to 30 meters at the top. So, over 100 meters, the radius decreases by 20 meters. So, the rate of change of radius per meter is -20/100 = -0.2 meters per meter.Therefore, at a height of 40 meters from the base, the radius would be 50 - 0.2*40 = 50 - 8 = 42 meters.Wait, that's conflicting with the previous answer of 38 meters. Hmm, now I'm confused.Wait, maybe I messed up the reference point. If I consider the truncated cone as a frustum, then the radius decreases from 50 meters at the bottom to 30 meters at the top over a height of 100 meters. So, the slope is (30 - 50)/100 = -0.2 per meter. So, starting from the bottom, at height 0, radius is 50 meters. At height 40 meters, the radius is 50 + (-0.2)*40 = 50 - 8 = 42 meters.But earlier, using the similar triangles approach, I got 38 meters. So, which one is correct?Wait, perhaps the confusion is about the reference point. In the similar triangles approach, I considered the entire original cone of height 250 meters, so the radius at 190 meters from the apex is 38 meters. But if we consider the truncated cone as starting at 150 meters from the apex, then the height from the base is 40 meters, which is 150 + 40 = 190 meters from the apex, so 38 meters is correct.But when I considered the frustum itself, with height 100 meters, the radius at 40 meters from the base would be 50 - 0.2*40 = 42 meters. So, which is correct?Wait, maybe I made a mistake in the similar triangles approach. Let me clarify.If the original cone is 250 meters tall with a base radius of 50 meters, then the radius at any height y from the apex is (y/250)*50.So, at y = 150 meters (the top of the truncated cone), the radius is (150/250)*50 = (3/5)*50 = 30 meters, which is correct.Similarly, at y = 250 meters, the radius is 50 meters.So, if we take a cross-section at 40 meters from the base of the truncated cone, that is at y = 150 + 40 = 190 meters from the apex.Radius at y = 190 meters is (190/250)*50 = (0.76)*50 = 38 meters.But when I considered the frustum itself, I thought of it as a separate entity with its own height and radii, so the radius at 40 meters from the base would be 50 - (20/100)*40 = 50 - 8 = 42 meters.Wait, so which is correct? Is it 38 or 42 meters?I think the confusion arises from whether we're measuring from the apex or from the base of the frustum.In the problem statement, it says the cross-section is taken at a height of 40 meters from the base. So, the base is the larger radius of 50 meters, and moving up 40 meters from there.So, in the frustum, the radius decreases from 50 meters at the base to 30 meters at the top, which is 100 meters up. So, over 100 meters, it decreases by 20 meters, so 0.2 meters per meter.Therefore, at 40 meters from the base, the radius should be 50 - 0.2*40 = 42 meters.But according to the similar triangles approach, considering the entire original cone, the radius is 38 meters.So, which one is correct?Wait, perhaps the similar triangles approach is wrong because the frustum is not a cone but a part of a cone. So, when we take a cross-section at 40 meters from the base of the frustum, it's equivalent to 190 meters from the apex, so the radius is 38 meters.But if we model the frustum as a separate entity, the radius decreases linearly from 50 to 30 over 100 meters, so at 40 meters, it's 42 meters.Wait, this is conflicting. Maybe I need to reconcile these two approaches.Let me think about it. If the original cone is 250 meters tall, with radius 50 meters at the base. Then, the frustum is the part from 150 meters to 250 meters. So, at 150 meters, the radius is 30 meters, and at 250 meters, it's 50 meters.So, the slope of the radius with respect to height is (50 - 30)/(250 - 150) = 20/100 = 0.2 meters per meter.So, from the apex, the radius increases by 0.2 meters per meter.Therefore, at 190 meters from the apex, the radius is 0.2*190 = 38 meters.But from the base of the frustum, which is at 150 meters from the apex, moving up 40 meters, so 190 meters from the apex, the radius is 38 meters.Alternatively, if we model the frustum as a separate entity, starting at radius 50 meters and decreasing to 30 meters over 100 meters, then the radius at 40 meters from the base would be 50 - (20/100)*40 = 42 meters.Wait, so which one is correct? It seems like the two methods are giving different results.Wait, perhaps the issue is that in the similar triangles approach, the radius is increasing with height from the apex, whereas in the frustum approach, the radius is decreasing with height from the base.So, if we take the frustum's base as the reference point, moving up 40 meters, the radius decreases by 0.2*40 = 8 meters, so 50 - 8 = 42 meters.But according to the original cone, the radius at 190 meters from the apex is 38 meters.Wait, but 190 meters from the apex is 60 meters below the top of the original cone, which is at 250 meters. So, 250 - 190 = 60 meters. So, the radius at 60 meters from the top would be 50 - 0.2*60 = 50 - 12 = 38 meters. That matches the similar triangles approach.Wait, so if we consider the frustum as part of the original cone, then at 40 meters from the base of the frustum (which is 150 meters from the apex), moving up 40 meters (to 190 meters from the apex), the radius is 38 meters.But if we model the frustum as a separate entity, with its own coordinate system, starting at 50 meters radius and decreasing to 30 meters over 100 meters, then at 40 meters from the base, the radius is 42 meters.So, which one is correct? The problem says it's a truncated cone, so it's a frustum. So, perhaps the correct approach is to model it as a frustum and calculate the radius accordingly.Wait, but the problem also mentions that it's a truncated cone, so it's part of a larger cone. So, perhaps the similar triangles approach is the correct one.Wait, but in the problem statement, it's a truncated cone with height 100 meters, top radius 30, bottom radius 50. So, it's a frustum. So, in that case, the radius at any height within the frustum can be calculated based on the linear decrease from 50 to 30 over 100 meters.So, perhaps the correct radius at 40 meters from the base is 42 meters.But then, why does the similar triangles approach give 38 meters? Because in that approach, we're considering the entire original cone, which is 250 meters tall, so the radius at 190 meters from the apex is 38 meters.Wait, but 190 meters from the apex is 60 meters from the top of the original cone, so the radius there would be 50 - 0.2*60 = 38 meters.So, both approaches are correct, but they are measuring from different reference points.In the problem, the cross-section is taken at a height of 40 meters from the base of the truncated cone, which is the frustum. So, the base of the frustum is at the bottom, radius 50 meters, and moving up 40 meters, the radius decreases.So, in the frustum's coordinate system, the radius at 40 meters from the base is 42 meters.But in the original cone's coordinate system, the radius at 190 meters from the apex is 38 meters.So, which one is the correct answer? The problem says "a cross-section of the cone at a height of 40 meters from the base." So, the base is the bottom of the truncated cone, so 40 meters up from there.Therefore, in the frustum's own terms, the radius at 40 meters from the base is 42 meters.But wait, let me think again. If the frustum is part of the original cone, then the radius at 40 meters from the base of the frustum (which is 150 meters from the apex) is 38 meters.But if we consider the frustum as a separate entity, the radius at 40 meters from its base is 42 meters.So, which interpretation is correct?I think the problem is referring to the frustum as a separate entity, so the radius at 40 meters from its base is 42 meters.But to be sure, let's think about the linear change in radius.In the frustum, the radius changes from 50 meters at the base to 30 meters at the top, over a height of 100 meters. So, the rate of change is (30 - 50)/100 = -0.2 meters per meter.So, at a height h from the base, the radius r is:r = 50 + (-0.2)*hSo, at h = 40 meters,r = 50 - 0.2*40 = 50 - 8 = 42 meters.Therefore, the radius is 42 meters.Wait, but earlier, using the similar triangles approach, I got 38 meters. So, which one is correct?I think the confusion is because the similar triangles approach is considering the entire original cone, whereas the frustum is a separate entity. So, if we model the frustum as a separate entity, the radius at 40 meters from its base is 42 meters.But if we consider it as part of the original cone, the radius at 190 meters from the apex is 38 meters.But the problem says "a cross-section of the cone at a height of 40 meters from the base." So, the base is the bottom of the frustum, so 40 meters up from there, so in the frustum's own terms, the radius is 42 meters.Therefore, I think the correct radius is 42 meters.Wait, but let me check this with another method.The formula for the radius at a certain height in a frustum is:r = R - ( (R - r) / H ) * hWhere R is the bottom radius, r is the top radius, H is the height of the frustum, and h is the height from the base.So, plugging in the numbers:R = 50, r = 30, H = 100, h = 40r = 50 - ( (50 - 30) / 100 ) * 40r = 50 - (20/100)*40r = 50 - 0.2*40r = 50 - 8 = 42 meters.Yes, that confirms it. So, the radius at 40 meters from the base is 42 meters.Wait, but earlier, using the similar triangles approach, I got 38 meters. So, why the discrepancy?Because in the similar triangles approach, we're considering the entire original cone, which is 250 meters tall, so the radius at 190 meters from the apex is 38 meters. But in the frustum's own coordinate system, the radius at 40 meters from its base is 42 meters.So, which one is the correct answer? The problem says "a cross-section of the cone at a height of 40 meters from the base." So, the base is the bottom of the frustum, so 40 meters up from there, so in the frustum's own terms, the radius is 42 meters.Therefore, the radius is 42 meters, and the area is π*(42)^2.But let me just make sure. If we model the frustum as a separate entity, the radius at 40 meters from the base is 42 meters. If we model it as part of the original cone, the radius at 190 meters from the apex is 38 meters. But since the problem refers to the height from the base of the frustum, the correct radius is 42 meters.Okay, so moving on. The area of the cross-section is π*r², so π*(42)^2.Calculating that:42^2 = 1764So, area = 1764π square meters.Now, the second part: calculating the volume of the truncated cone portion above the cross-section at 40 meters from the base.So, the original frustum has a height of 100 meters, with radii 50 and 30. We need to find the volume above the cross-section at 40 meters from the base, which is a smaller frustum with height 60 meters (since 100 - 40 = 60), with radii 42 meters (at the base of this smaller frustum) and 30 meters (at the top).Wait, no. Wait, the cross-section is at 40 meters from the base of the original frustum, so the remaining part above it is a frustum with height 60 meters, with radii 42 meters (at the base of this upper frustum) and 30 meters (at the top).Alternatively, we can think of it as a smaller frustum with height 60 meters, top radius 30 meters, and bottom radius 42 meters.The formula for the volume of a frustum is:V = (1/3)πh(R² + Rr + r²)Where h is the height, R is the bottom radius, and r is the top radius.So, plugging in the numbers:h = 60 metersR = 42 metersr = 30 metersSo,V = (1/3)π*60*(42² + 42*30 + 30²)First, calculate 42² = 176442*30 = 126030² = 900So, summing these up:1764 + 1260 + 900 = 1764 + 1260 = 3024; 3024 + 900 = 3924So,V = (1/3)π*60*3924Simplify:(1/3)*60 = 20So,V = 20π*3924Calculating 20*3924:20*3924 = 78,480So, V = 78,480π cubic meters.Alternatively, we can think of it as subtracting the volume below the cross-section from the total volume of the frustum.The total volume of the original frustum is:V_total = (1/3)π*100*(50² + 50*30 + 30²)Calculating that:50² = 250050*30 = 150030² = 900Sum: 2500 + 1500 + 900 = 4900So,V_total = (1/3)π*100*4900 = (1/3)*100*4900π = (100/3)*4900π ≈ 33,333.333π * 4900? Wait, no.Wait, (1/3)*100 = 100/3 ≈ 33.333So, 33.333 * 4900 ≈ 163,333.333π cubic meters.Wait, but that's the total volume. Now, the volume below the cross-section at 40 meters is a frustum with height 40 meters, radii 50 meters and 42 meters.So, V_below = (1/3)π*40*(50² + 50*42 + 42²)Calculating:50² = 250050*42 = 210042² = 1764Sum: 2500 + 2100 + 1764 = 2500 + 2100 = 4600; 4600 + 1764 = 6364So,V_below = (1/3)π*40*6364Simplify:(1/3)*40 = 40/3 ≈ 13.333So,V_below = 13.333π*6364 ≈ 13.333*6364π ≈ 84,853.333π cubic meters.Wait, but the total volume was approximately 163,333.333π, so subtracting V_below from V_total would give V_above = 163,333.333π - 84,853.333π ≈ 78,480π cubic meters, which matches the earlier calculation.Therefore, the volume above the cross-section is 78,480π cubic meters.Alternatively, we can use the similar cones approach. Since the original frustum is part of a larger cone of height 250 meters, the volume above the cross-section can be found by subtracting the volume of the smaller cone (up to 190 meters) from the volume of the larger cone (up to 250 meters).Wait, let me try that.The volume of the original cone (height 250 meters, radius 50 meters) is:V_original = (1/3)π*(50)^2*250 = (1/3)π*2500*250 = (1/3)*625,000π ≈ 208,333.333π cubic meters.The volume of the smaller cone (height 190 meters, radius 38 meters) is:V_smaller = (1/3)π*(38)^2*190Calculating:38² = 1,4441,444 * 190 = Let's compute 1,444 * 200 = 288,800; subtract 1,444*10 = 14,440, so 288,800 - 14,440 = 274,360So,V_smaller = (1/3)π*274,360 ≈ 91,453.333π cubic meters.Therefore, the volume of the frustum above the cross-section is V_original - V_smaller = 208,333.333π - 91,453.333π ≈ 116,880π cubic meters.Wait, that's different from the 78,480π we got earlier. So, which one is correct?Wait, no, because the frustum above the cross-section is not the same as the difference between the original cone and the smaller cone. Because the frustum above the cross-section is part of the original frustum, which itself is part of the original cone.Wait, actually, the frustum above the cross-section is a frustum of the original cone, with height 60 meters (from 190 to 250 meters), radii 38 meters and 50 meters.Wait, no, at 190 meters, the radius is 38 meters, and at 250 meters, it's 50 meters. So, the frustum above the cross-section is a frustum of the original cone with height 60 meters, radii 38 meters and 50 meters.So, its volume would be:V_above = (1/3)π*60*(38² + 38*50 + 50²)Calculating:38² = 1,44438*50 = 1,90050² = 2,500Sum: 1,444 + 1,900 + 2,500 = 1,444 + 1,900 = 3,344; 3,344 + 2,500 = 5,844So,V_above = (1/3)π*60*5,844Simplify:(1/3)*60 = 20So,V_above = 20π*5,844 = 116,880π cubic meters.But earlier, using the frustum approach, we got 78,480π cubic meters.So, which one is correct?Wait, I think the confusion is that the frustum above the cross-section is part of the original cone, but the cross-section is at 40 meters from the base of the original frustum, which is itself part of the original cone.Wait, perhaps the correct approach is to model the frustum above the cross-section as a frustum with height 60 meters, radii 42 meters and 30 meters, which gives 78,480π cubic meters.But according to the similar cones approach, considering the original cone, the volume above the cross-section is 116,880π cubic meters.So, which one is correct?Wait, perhaps the problem is referring to the volume above the cross-section within the original frustum, not within the original cone.So, the original frustum has a volume of 163,333.333π cubic meters. The volume below the cross-section is 84,853.333π cubic meters, so the volume above is 78,480π cubic meters.Therefore, the correct volume is 78,480π cubic meters.Alternatively, if we consider the frustum above the cross-section as part of the original cone, it's a frustum with radii 38 meters and 50 meters, height 60 meters, giving 116,880π cubic meters. But that's the volume of that frustum within the original cone, not within the original frustum.So, since the problem is about the geological formation, which is the original frustum, the volume above the cross-section is 78,480π cubic meters.Therefore, the answers are:1. Radius = 42 meters, Area = 1764π square meters.2. Volume = 78,480π cubic meters.But let me just make sure.Wait, in the first part, I got 42 meters using the frustum approach, and 38 meters using the similar cones approach. But since the problem refers to the height from the base of the frustum, the correct radius is 42 meters.Similarly, for the volume, since we're dealing with the frustum, the volume above the cross-section is 78,480π cubic meters.Therefore, the final answers are:1. Radius = 42 meters, Area = 1764π m².2. Volume = 78,480π m³.But let me just check the volume formula again.The formula for the volume of a frustum is:V = (1/3)πh(R² + Rr + r²)So, for the frustum above the cross-section, h = 60 meters, R = 42 meters, r = 30 meters.So,V = (1/3)π*60*(42² + 42*30 + 30²)Calculating:42² = 176442*30 = 126030² = 900Sum: 1764 + 1260 + 900 = 3924So,V = (1/3)*π*60*3924 = 20π*3924 = 78,480π cubic meters.Yes, that's correct.Therefore, the answers are:1. Radius = 42 meters, Area = 1764π m².2. Volume = 78,480π m³.</think>