🧠:<think>Alright, so I've got this problem about an automotive designer optimizing a car's surface for both aerodynamic efficiency and structural precision. It's divided into two parts, and I need to tackle each one step by step. Let me start by understanding what each part is asking.Problem 1: Average Curvature for Aerodynamic EfficiencyThe first part is about calculating the average curvature ( H ) of the car's surface over a specific region. The surface is defined by ( f(x, y, z) = 0 ), where ( f ) is a polynomial of degree 3. They also mention that ( f(x, y, z) = z - g(x, y) ), with ( g ) being a polynomial of degree 3. The region of interest is ( x^2 + y^2 leq R^2 ), which is a circle in the xy-plane with radius ( R ).I remember that curvature for surfaces can be a bit tricky. There are different types, like Gaussian curvature and mean curvature. Since the problem mentions the average curvature, I think they're referring to the mean curvature ( H ). Mean curvature is the average of the principal curvatures at a given point on the surface.To find the average curvature over a region, I need to integrate the mean curvature over that region and then divide by the area of the region. So, the average curvature ( bar{H} ) would be:[bar{H} = frac{1}{text{Area}} iint_{x^2 + y^2 leq R^2} H(x, y) , dx , dy]But first, I need to find the mean curvature ( H ) of the surface ( f(x, y, z) = 0 ). Since the surface is given implicitly, I can use the formula for mean curvature for an implicit surface.The general formula for mean curvature ( H ) for a surface ( F(x, y, z) = 0 ) is:[H = frac{F_x F_{yy} + F_y F_{xy} - F_x F_{xz} F_z + F_y F_{xz} F_x - F_z F_{xy} F_y + F_z^2 F_{xx} + F_z^2 F_{yy} - F_z F_{xz} F_x - F_z F_{yz} F_y}{(F_x^2 + F_y^2 + F_z^2)^{3/2}}]Wait, that seems complicated. Maybe there's a simpler way since the surface is given as ( z = g(x, y) ). If I can express the surface explicitly as ( z = g(x, y) ), then the mean curvature can be computed more straightforwardly.Yes, that's right. For a surface given by ( z = g(x, y) ), the mean curvature ( H ) is:[H = frac{g_{xx} + g_{yy}}{2 left(1 + g_x^2 + g_y^2right)^{3/2}}]Where ( g_x ) and ( g_y ) are the first partial derivatives, and ( g_{xx} ), ( g_{yy} ) are the second partial derivatives.So, since ( f(x, y, z) = z - g(x, y) = 0 ), we can write ( z = g(x, y) ). Therefore, the mean curvature is as above.Now, to compute the average curvature over the region ( x^2 + y^2 leq R^2 ), I need to set up the integral:[bar{H} = frac{1}{pi R^2} iint_{x^2 + y^2 leq R^2} frac{g_{xx} + g_{yy}}{2 left(1 + g_x^2 + g_y^2right)^{3/2}} , dx , dy]But wait, ( g(x, y) ) is a polynomial of degree 3. That might complicate things because the integral could be quite involved. Maybe there's a way to simplify this expression or find symmetries.Alternatively, perhaps the average curvature can be related to some integral of the Laplacian of ( g ), since ( g_{xx} + g_{yy} ) is the Laplacian ( Delta g ). So, the numerator is ( Delta g / 2 ), and the denominator is ( (1 + |nabla g|^2)^{3/2} ).Hmm, integrating this over a circular region. Since ( g ) is a cubic polynomial, its derivatives will be quadratic and linear. Let me think about the properties of such a function.A cubic polynomial in two variables can be written as:[g(x, y) = ax^3 + by^3 + cx^2 y + dx y^2 + ex^2 + f y^2 + gx y + hx + iy + j]But since ( f(x, y, z) = z - g(x, y) ) is a polynomial of degree 3, ( g ) must also be degree 3. So, the terms above are all possible up to degree 3.Now, when we take the Laplacian ( Delta g = g_{xx} + g_{yy} ), we get:[Delta g = 6a x + 6b y + 2c y + 2d x + 2e + 2f]Wait, let me compute that properly.For ( g(x, y) = ax^3 + by^3 + cx^2 y + dx y^2 + ex^2 + f y^2 + gx y + hx + iy + j ):- ( g_x = 3a x^2 + 2c x y + 2e x + g y + h )- ( g_y = 3b y^2 + c x^2 + 2d x y + 2f y + g x + i )- ( g_{xx} = 6a x + 2c y + 2e )- ( g_{yy} = 6b y + 2c x + 2f )- So, ( Delta g = g_{xx} + g_{yy} = 6a x + 6b y + 2c x + 2c y + 2e + 2f )Simplify:[Delta g = (6a + 2c) x + (6b + 2c) y + (2e + 2f)]So, ( Delta g ) is a linear function in ( x ) and ( y ).Similarly, the denominator ( (1 + g_x^2 + g_y^2)^{3/2} ) is more complicated because ( g_x ) and ( g_y ) are quadratic. So, ( g_x^2 + g_y^2 ) will be quartic, and raising that to the 3/2 power will make it a degree 6 function.But integrating this over a circular region might be challenging. However, perhaps due to the symmetry of the region ( x^2 + y^2 leq R^2 ), some terms might integrate to zero.Let me consider the integral:[iint_{x^2 + y^2 leq R^2} frac{Delta g}{2 (1 + g_x^2 + g_y^2)^{3/2}} , dx , dy]Given that ( Delta g ) is linear in ( x ) and ( y ), and the denominator is a function that's even in ( x ) and ( y ), the integrand will be an odd function in ( x ) and ( y ) if ( Delta g ) is linear.Wait, let's think about this. If we have a function that is linear in ( x ) and ( y ), multiplied by a function that's even in ( x ) and ( y ), then the product will be odd in ( x ) and ( y ). Because linear terms are odd functions, and even functions multiplied by odd functions give odd functions.Therefore, integrating an odd function over a symmetric region like a circle centered at the origin will result in zero. Because for every point ( (x, y) ), there's a corresponding point ( (-x, -y) ) that cancels out the contribution.So, does that mean the integral of ( Delta g ) over the region, divided by the denominator, is zero?Wait, but ( Delta g ) is linear, and the denominator is even, so the integrand is odd. Therefore, the integral over the symmetric region is zero.Therefore, the average curvature ( bar{H} ) would be zero?But that seems counterintuitive. How can the average curvature be zero? Maybe I made a mistake.Wait, let me double-check. The mean curvature is ( frac{Delta g}{2 (1 + |nabla g|^2)^{3/2}} ). So, the numerator is linear, denominator is even. So, the integrand is odd.But integrating an odd function over a symmetric interval gives zero. So, the average curvature would indeed be zero.But wait, is that possible? For a surface defined by a cubic polynomial, could the average curvature over a circular region be zero?Alternatively, maybe I'm missing something. Perhaps the mean curvature isn't just ( Delta g / (2 (1 + |nabla g|^2)^{3/2}) ). Let me verify the formula.Yes, for a surface ( z = g(x, y) ), the mean curvature is indeed:[H = frac{g_{xx} + g_{yy}}{2 left(1 + g_x^2 + g_y^2right)^{3/2}}]So, that part is correct.Therefore, since ( Delta g ) is linear, and the denominator is even, the integrand is odd, so the integral over the symmetric region is zero. Hence, the average curvature ( bar{H} = 0 ).But wait, that seems too straightforward. Maybe I need to consider the actual expression more carefully.Alternatively, perhaps the average curvature isn't zero because the denominator isn't exactly even, but depends on ( g_x ) and ( g_y ), which are quadratic. However, ( g_x ) and ( g_y ) are quadratic, so their squares are quartic, and the denominator is ( (1 + text{quartic})^{3/2} ), which is still even in ( x ) and ( y ).Therefore, the denominator is even, and the numerator is linear (odd), so the product is odd. Therefore, integrating over a symmetric region gives zero.So, the average curvature ( bar{H} = 0 ).But wait, let me think again. If the surface is symmetric, maybe the mean curvature is symmetric in such a way that the average is zero. But is that always the case?Alternatively, perhaps the cubic terms in ( g ) lead to certain symmetries that make the integral zero.Alternatively, maybe I need to consider that ( g ) is a cubic polynomial, so ( Delta g ) is linear, and when integrated against an even function, it cancels out.Yes, that seems consistent.Therefore, the average curvature ( bar{H} = 0 ).But wait, is that possible? For example, consider a simple cubic function like ( g(x, y) = x^3 ). Then, ( Delta g = 6x ), which is linear. The denominator would be ( (1 + (3x^2)^2 + (0)^2)^{3/2} = (1 + 9x^4)^{3/2} ), which is even. So, the integrand is ( 6x / (2 (1 + 9x^4)^{3/2}) ), which is an odd function. Therefore, integrating over symmetric limits (e.g., from -R to R in x, and similarly in y) would give zero.Therefore, yes, the average curvature would be zero.So, for any cubic polynomial ( g(x, y) ), the average curvature over the symmetric region ( x^2 + y^2 leq R^2 ) is zero.Therefore, the answer to part 1 is that the average curvature ( H ) is zero.Problem 2: Stress Tensor ConstraintNow, moving on to the second part. The designer wants the stress tensor ( sigma ) to satisfy ( nabla cdot sigma = mathbf{0} ) everywhere on the surface. The stress tensor is given as the Hessian of ( f ), i.e., ( sigma_{ij} = frac{partial^2 f}{partial x_i partial x_j} ).So, ( sigma ) is a symmetric tensor because the Hessian is symmetric. The condition ( nabla cdot sigma = mathbf{0} ) means that the divergence of the stress tensor is zero vector.In index notation, the divergence of a tensor ( sigma ) is given by:[(nabla cdot sigma)_i = frac{partial sigma_{ij}}{partial x_j}]So, for each component ( i ), we sum over ( j ) the partial derivatives of ( sigma_{ij} ) with respect to ( x_j ).Given that ( sigma_{ij} = frac{partial^2 f}{partial x_i partial x_j} ), then:[(nabla cdot sigma)_i = frac{partial}{partial x_j} left( frac{partial^2 f}{partial x_i partial x_j} right ) = frac{partial^3 f}{partial x_j partial x_i partial x_j}]Wait, but the indices are summed over, so it's:[(nabla cdot sigma)_i = sum_{j=1}^3 frac{partial^3 f}{partial x_j partial x_i partial x_j}]But since mixed partial derivatives are equal ( Clairaut's theorem ), we can rearrange the derivatives:[frac{partial^3 f}{partial x_j partial x_i partial x_j} = frac{partial^3 f}{partial x_i partial x_j partial x_j} = frac{partial}{partial x_i} left( frac{partial^2 f}{partial x_j^2} right )]Therefore, the divergence condition becomes:[sum_{j=1}^3 frac{partial}{partial x_i} left( frac{partial^2 f}{partial x_j^2} right ) = 0 quad text{for each } i = 1, 2, 3]Which can be written as:[frac{partial}{partial x_i} left( sum_{j=1}^3 frac{partial^2 f}{partial x_j^2} right ) = 0]But ( sum_{j=1}^3 frac{partial^2 f}{partial x_j^2} ) is the Laplacian of ( f ), denoted as ( Delta f ). So, the condition becomes:[frac{partial (Delta f)}{partial x_i} = 0 quad text{for each } i = 1, 2, 3]This implies that the Laplacian of ( f ) must be a constant function. Because if the partial derivatives of ( Delta f ) with respect to each variable are zero, then ( Delta f ) doesn't change with any variable, so it's constant.Therefore, ( Delta f = C ), where ( C ) is a constant.But ( f ) is a polynomial of degree 3. Let's consider the Laplacian of a cubic polynomial.A general cubic polynomial in three variables is:[f(x, y, z) = sum_{i+j+k leq 3} a_{ijk} x^i y^j z^k]The Laplacian ( Delta f ) is:[Delta f = f_{xx} + f_{yy} + f_{zz}]Each second derivative will reduce the degree by 2. So, the Laplacian of a cubic polynomial is a linear function (degree 1).But we have ( Delta f = C ), a constant. Therefore, the Laplacian must be a constant, which is a degree 0 polynomial.Therefore, the coefficients of all terms of degree 1 in ( Delta f ) must be zero.Let me compute ( Delta f ) for a general cubic polynomial.Let me write ( f(x, y, z) ) as:[f(x, y, z) = a x^3 + b y^3 + c z^3 + d x^2 y + e x^2 z + f y^2 x + g y^2 z + h z^2 x + i z^2 y + j x y z + k x^2 + l y^2 + m z^2 + n x y + o x z + p y z + q x + r y + s z + t]Now, compute the second derivatives:- ( f_{xx} = 6a x + 2d y + 2e z + 2k )- ( f_{yy} = 6b y + 2d x + 2f x + 2g z + 2l )- ( f_{zz} = 6c z + 2e x + 2g y + 2h x + 2i y + 2m )Wait, let me do this step by step.Compute ( f_{xx} ):- The term ( a x^3 ) gives ( 6a x )- The term ( d x^2 y ) gives ( 2d y )- The term ( e x^2 z ) gives ( 2e z )- The term ( k x^2 ) gives ( 2k )- All other terms either don't have ( x^2 ) or their second derivative with respect to x is zero.Similarly, ( f_{yy} ):- The term ( b y^3 ) gives ( 6b y )- The term ( d x^2 y ) gives ( 2d x )- The term ( f y^2 x ) gives ( 2f x )- The term ( g y^2 z ) gives ( 2g z )- The term ( l y^2 ) gives ( 2l )- All other terms either don't have ( y^2 ) or their second derivative with respect to y is zero.Similarly, ( f_{zz} ):- The term ( c z^3 ) gives ( 6c z )- The term ( e x^2 z ) gives ( 2e x )- The term ( g y^2 z ) gives ( 2g y )- The term ( h z^2 x ) gives ( 2h x )- The term ( i z^2 y ) gives ( 2i y )- The term ( m z^2 ) gives ( 2m )- All other terms either don't have ( z^2 ) or their second derivative with respect to z is zero.Now, summing these up for ( Delta f = f_{xx} + f_{yy} + f_{zz} ):[Delta f = (6a x + 2d y + 2e z + 2k) + (6b y + 2d x + 2f x + 2g z + 2l) + (6c z + 2e x + 2g y + 2h x + 2i y + 2m)]Now, let's collect like terms:- Terms with ( x ): - ( 6a x ) - ( 2d x ) - ( 2f x ) - ( 2e x ) - ( 2h x ) Total: ( (6a + 2d + 2f + 2e + 2h) x )- Terms with ( y ): - ( 2d y ) - ( 6b y ) - ( 2g y ) - ( 2i y ) Total: ( (2d + 6b + 2g + 2i) y )- Terms with ( z ): - ( 2e z ) - ( 2g z ) - ( 6c z ) Total: ( (2e + 2g + 6c) z )- Constant terms: - ( 2k ) - ( 2l ) - ( 2m ) Total: ( 2(k + l + m) )So, putting it all together:[Delta f = [6a + 2d + 2f + 2e + 2h] x + [2d + 6b + 2g + 2i] y + [2e + 2g + 6c] z + 2(k + l + m)]But we have the condition that ( Delta f = C ), a constant. Therefore, the coefficients of ( x ), ( y ), and ( z ) must be zero, and the constant term is ( C ).Therefore, we have the following system of equations:1. Coefficient of ( x ): ( 6a + 2d + 2f + 2e + 2h = 0 )2. Coefficient of ( y ): ( 2d + 6b + 2g + 2i = 0 )3. Coefficient of ( z ): ( 2e + 2g + 6c = 0 )4. Constant term: ( 2(k + l + m) = C )So, these are the conditions that the coefficients of ( f ) must satisfy.But since ( f ) is a polynomial of degree 3, and we're looking for conditions on ( f ), we can express these as constraints on the coefficients.Therefore, the conditions on ( f ) are:- ( 6a + 2d + 2f + 2e + 2h = 0 )- ( 2d + 6b + 2g + 2i = 0 )- ( 2e + 2g + 6c = 0 )- ( k + l + m = C/2 )But since ( C ) is a constant, we can set ( C ) to any value, but typically in such problems, unless specified, we might assume ( C = 0 ) for simplicity, but the problem doesn't specify, so we can just leave it as ( k + l + m = C/2 ).Therefore, the conditions on ( f ) are the above four equations.But let me check if I did the coefficients correctly.Wait, in the term for ( f_{yy} ), I had ( 2f x ) from ( f y^2 x ). Similarly, in ( f_{zz} ), I had ( 2e x ) from ( e x^2 z ), and ( 2h x ) from ( h z^2 x ). So, yes, the coefficients for ( x ) are correctly collected.Similarly, for ( y ), from ( f_{xx} ) we have ( 2d y ), from ( f_{yy} ) we have ( 6b y ), ( 2g y ), and from ( f_{zz} ) we have ( 2i y ). So, that's correct.For ( z ), from ( f_{xx} ) we have ( 2e z ), from ( f_{yy} ) we have ( 2g z ), and from ( f_{zz} ) we have ( 6c z ). Correct.And the constants are ( 2k + 2l + 2m ), which is ( 2(k + l + m) ). So, correct.Therefore, the conditions on ( f ) are:1. ( 6a + 2d + 2e + 2f + 2h = 0 )2. ( 2d + 6b + 2g + 2i = 0 )3. ( 2e + 2g + 6c = 0 )4. ( k + l + m = C/2 )Where ( C ) is a constant.But since ( f ) is a polynomial of degree 3, and we're looking for the conditions on ( f ), these are the necessary conditions.Alternatively, we can write these conditions in terms of the coefficients of ( f ).Therefore, the conditions are that the coefficients satisfy the above four equations.So, summarizing, for the stress tensor condition ( nabla cdot sigma = 0 ), the polynomial ( f ) must satisfy that its Laplacian is a constant, which imposes the four conditions above on its coefficients.Final Answer1. The average curvature ( H ) is (boxed{0}).2. The conditions on ( f ) are that the coefficients satisfy: [ begin{cases} 6a + 2d + 2e + 2f + 2h = 0, 2d + 6b + 2g + 2i = 0, 2e + 2g + 6c = 0, k + l + m = dfrac{C}{2}, end{cases} ] where ( C ) is a constant. Therefore, the conditions are (boxed{6a + 2d + 2e + 2f + 2h = 0}), (boxed{2d + 6b + 2g + 2i = 0}), (boxed{2e + 2g + 6c = 0}), and (boxed{k + l + m = dfrac{C}{2}}).</think>

🧠:<think>Alright, so I have this problem about Alex, a philanthropist who donates a total of 1,000,000 to three causes: education, healthcare, and environmental conservation. There are some specific conditions given, and I need to figure out how much Alex donates to each cause. Then, there's also a part about investing a portion of the total donation, which appreciates at 5% annually, and ensuring that the value at the end of the year is at least 210,000. Hmm, okay, let me break this down step by step.First, let's focus on the donations. There are three causes, and the total donation is 1,000,000. The conditions are:1. The amount donated to education is twice the amount donated to healthcare.2. The amount donated to environmental conservation is 150,000 more than the amount donated to healthcare.So, let me assign variables to each cause to make it easier. Let's let H represent the amount donated to healthcare. Then, according to condition 1, the amount donated to education would be 2H. And according to condition 2, the amount donated to environmental conservation would be H + 150,000.So, in terms of equations, that's:Education: 2HHealthcare: HEnvironmental Conservation: H + 150,000Since the total donation is 1,000,000, we can write the equation:Education + Healthcare + Environmental Conservation = 1,000,000Substituting the expressions in terms of H:2H + H + (H + 150,000) = 1,000,000Let me simplify that:2H + H + H + 150,000 = 1,000,000Combining like terms:2H + H + H is 4H, so:4H + 150,000 = 1,000,000Now, subtract 150,000 from both sides to solve for H:4H = 1,000,000 - 150,0004H = 850,000Then, divide both sides by 4:H = 850,000 / 4Let me compute that. 850,000 divided by 4. 4 goes into 8 two times, 4 into 5 once with remainder 1, 4 into 10 twice with remainder 2, 4 into 20 five times. Wait, maybe I should do it more carefully.850,000 divided by 4:4 into 8 is 2, 4 into 5 is 1 with 1 remainder, bring down the 0: 10. 4 into 10 is 2, remainder 2. Bring down the 0: 20. 4 into 20 is 5. Bring down the next 0: 0. 4 into 0 is 0. Bring down the last 0: 0. So, putting it all together: 212,500.So, H = 212,500.Therefore, the amount donated to healthcare is 212,500.Then, the amount donated to education is twice that, so 2 * 212,500 = 425,000.And the amount donated to environmental conservation is H + 150,000, which is 212,500 + 150,000 = 362,500.Let me double-check that these add up to 1,000,000:212,500 (Healthcare) + 425,000 (Education) + 362,500 (Environmental) = ?212,500 + 425,000 is 637,500. Then, 637,500 + 362,500 is 1,000,000. Perfect, that checks out.So, part 1 is done. Now, moving on to part 2. Alex decides to invest a portion of the total donation amount in a fund that appreciates annually at a rate of 5%. At the end of the year, the value of this investment must be at least 210,000.So, we need to find the initial amount Alex invests, let's call it P, such that after one year, with 5% interest, it becomes at least 210,000.The formula for the future value of an investment with simple interest is:Future Value = Principal * (1 + rate)So, in this case, the future value needs to be at least 210,000, and the rate is 5%, which is 0.05.So, the equation is:P * (1 + 0.05) ≥ 210,000Simplify 1 + 0.05 to 1.05:P * 1.05 ≥ 210,000To find P, we divide both sides by 1.05:P ≥ 210,000 / 1.05Let me compute that. 210,000 divided by 1.05.Well, 1.05 times 200,000 is 210,000 because 1.05 * 200,000 = 210,000. So, 210,000 / 1.05 = 200,000.Therefore, P must be at least 200,000.Wait, that seems straightforward. So, Alex needs to invest at least 200,000 initially to have at least 210,000 after one year with 5% appreciation.But hold on, let me verify that.If P is 200,000, then after 5% appreciation, it becomes 200,000 * 1.05 = 210,000. So, exactly 210,000. So, if Alex invests exactly 200,000, the future value is exactly 210,000. If he invests more, it would be more than 210,000. So, the minimal amount he needs to invest is 200,000.Therefore, the initial investment should be 200,000.But wait, hold on another thought. The problem says "a portion of the total donation amount." So, the total donation is 1,000,000, and Alex is investing a portion of that. So, is the investment part of the 1,000,000, or is it in addition to it? Hmm, the wording says "invests a portion of the total donation amount." So, it's a portion of the 1,000,000. So, the investment is part of the donations, meaning that the donations to the three causes plus the investment sum up to 1,000,000.Wait, no, actually, the problem says "Alex donates a total of 1,000,000" and "decides to invest a portion of the total donation amount in a fund." So, the total donation is 1,000,000, and from that, he sets aside some portion to invest. So, the donations to the three causes plus the investment equal 1,000,000.Wait, but in part 1, we already allocated the entire 1,000,000 to the three causes. So, now, in part 2, is he taking a portion from the total donation, meaning that the donations to the three causes would have to be adjusted? Or is the investment in addition to the donations?Wait, let me read the problem again."A philanthropist, Alex, donates to three different causes: education, healthcare, and environmental conservation. Over the course of a year, Alex donates a total of 1,000,000. The donations to each cause adhere to the following conditions... Additionally, Alex decides to invest a portion of the total donation amount in a fund that appreciates annually at a rate of 5%. At the end of the year, the value of this investment must be at least 210,000."So, the total donation is 1,000,000, which is donated to the three causes. Additionally, Alex invests a portion of the total donation amount. So, the investment is separate from the donations? Or is it part of the donations?Wait, that's a bit ambiguous. If the total donation is 1,000,000, and he invests a portion of that, then the investments are part of the donations. So, the donations to the three causes plus the investment equal 1,000,000.But in part 1, we already allocated the entire 1,000,000 to the three causes. So, perhaps the investment is in addition to the donations? That would make the total amount Alex spends 1,000,000 plus the investment. But the problem says "invests a portion of the total donation amount," which suggests that the investment is part of the 1,000,000.Hmm, this is a bit confusing. Let me read it again."Alex donates a total of 1,000,000. The donations to each cause adhere to the following conditions... Additionally, Alex decides to invest a portion of the total donation amount in a fund that appreciates annually at a rate of 5%. At the end of the year, the value of this investment must be at least 210,000."So, the total donation is 1,000,000, which is given to the three causes. Additionally, he invests a portion of that total donation amount. So, is the investment part of the 1,000,000? Or is it in addition?The wording is a bit unclear, but I think it's part of the 1,000,000. So, the 1,000,000 is divided into three parts: education, healthcare, environmental conservation, and an investment. So, the sum of these four should be 1,000,000.But in part 1, we only considered the three causes. So, perhaps part 1 is just about the donations, and part 2 is about the investment, which is separate.Wait, the problem says "Additionally, Alex decides to invest a portion of the total donation amount..." So, the total donation is 1,000,000, and he invests a portion of that. So, the investment is part of the 1,000,000. Therefore, the three causes plus the investment add up to 1,000,000.But in part 1, we already allocated the entire 1,000,000 to the three causes. So, perhaps the initial problem is that the three causes and the investment are all part of the 1,000,000. So, we need to adjust our earlier calculations.Wait, that complicates things. Let me see.Wait, the problem is structured as two separate questions:1. Determine how much Alex donates to each cause.2. Calculate the initial amount Alex invests in the fund to meet the appreciation requirement at the end of the year.So, perhaps part 1 is just about the donations, and part 2 is about the investment, which is in addition to the donations? But the wording says "invests a portion of the total donation amount," which is 1,000,000.Hmm, maybe the investment is part of the 1,000,000. So, the total amount is 1,000,000, which is split into three donations and the investment. So, the sum of the three donations plus the investment is 1,000,000.But in that case, part 1 would require us to find the donations, and part 2 the investment, with the total being 1,000,000.But in the initial problem, the conditions are only about the donations, not about the investment. So, perhaps part 1 is just about the donations, assuming that the investment is in addition to the 1,000,000. But that would make the total expenditure more than 1,000,000, which might not make sense.Wait, let's think again.The problem says: "Alex donates a total of 1,000,000." So, that's the total donation. Then, "Additionally, Alex decides to invest a portion of the total donation amount in a fund..." So, the investment is in addition to the donations? Or is it part of the donations?The phrase "a portion of the total donation amount" suggests that it's part of the 1,000,000. So, the 1,000,000 is split into three donations and the investment.Therefore, in part 1, we need to find the donations, considering that a portion is invested. So, perhaps the initial equations need to include the investment.Wait, but in the initial problem statement, the conditions are only about the donations, not about the investment. So, maybe part 1 is just about the donations, and part 2 is about the investment, which is separate.But the problem says "invests a portion of the total donation amount," which is 1,000,000. So, the investment is part of the 1,000,000. Therefore, the donations to the three causes plus the investment equal 1,000,000.Therefore, in part 1, when we calculated the donations, we didn't consider the investment, but actually, the investment is part of the 1,000,000, so we need to adjust our equations.Wait, that changes things. So, perhaps I need to redo part 1, considering that the total is 1,000,000, which includes the three donations and the investment.But the problem is structured as two separate questions: first, determine the donations, then calculate the investment. So, maybe part 1 is just about the donations, and part 2 is about the investment, which is part of the 1,000,000.Wait, this is getting a bit tangled. Let me try to parse the problem again."A philanthropist, Alex, donates to three different causes: education, healthcare, and environmental conservation. Over the course of a year, Alex donates a total of 1,000,000. The donations to each cause adhere to the following conditions... Additionally, Alex decides to invest a portion of the total donation amount in a fund that appreciates annually at a rate of 5%. At the end of the year, the value of this investment must be at least 210,000."So, the total donation is 1,000,000, which is given to the three causes. Additionally, he invests a portion of that total donation amount. So, the investment is part of the 1,000,000.Therefore, the three donations plus the investment equal 1,000,000.Therefore, in part 1, when I calculated the donations, I didn't account for the investment, which is part of the total. So, my initial calculation was incorrect because I assumed the entire 1,000,000 was donated, but actually, a portion is invested.So, let me correct that.Let me define variables again, but this time including the investment.Let H be the amount donated to healthcare.Then, education is 2H, environmental conservation is H + 150,000.Let I be the amount invested.So, total donation is H + 2H + (H + 150,000) + I = 1,000,000Simplify:H + 2H + H + 150,000 + I = 1,000,000Combine like terms:4H + 150,000 + I = 1,000,000So, 4H + I = 850,000But from part 2, we have another equation related to the investment.The investment I, when appreciated at 5%, must be at least 210,000.So, I * 1.05 ≥ 210,000Therefore, I ≥ 210,000 / 1.05 = 200,000So, I must be at least 200,000.Therefore, plugging back into the total equation:4H + I = 850,000Since I must be at least 200,000, the maximum amount that can be donated is when I is exactly 200,000.So, 4H + 200,000 = 850,000Therefore, 4H = 650,000H = 650,000 / 4 = 162,500So, H = 162,500Therefore, donations:Healthcare: 162,500Education: 2 * 162,500 = 325,000Environmental Conservation: 162,500 + 150,000 = 312,500Investment: 200,000Let me check the total:162,500 + 325,000 + 312,500 + 200,000 = ?162,500 + 325,000 = 487,500487,500 + 312,500 = 800,000800,000 + 200,000 = 1,000,000Perfect, that adds up.So, in this case, the donations are:Education: 325,000Healthcare: 162,500Environmental Conservation: 312,500And the investment is 200,000.But wait, this contradicts my initial thought where I didn't consider the investment as part of the total donation. So, now, I'm confused because the problem is structured as two separate questions. So, maybe the initial assumption was correct, that the investment is in addition to the donations.Wait, let me read the problem again."A philanthropist, Alex, donates to three different causes: education, healthcare, and environmental conservation. Over the course of a year, Alex donates a total of 1,000,000. The donations to each cause adhere to the following conditions... Additionally, Alex decides to invest a portion of the total donation amount in a fund that appreciates annually at a rate of 5%. At the end of the year, the value of this investment must be at least 210,000."So, the total donation is 1,000,000, which is given to the three causes. Additionally, he invests a portion of that total donation amount. So, the investment is part of the 1,000,000.Therefore, the three donations plus the investment equal 1,000,000.Therefore, in part 1, when determining the donations, we have to consider that a portion is invested, so the donations are less than 1,000,000.Therefore, the correct approach is to include the investment in the total.So, my initial calculation was wrong because I didn't consider the investment. So, the correct way is to set up the equations considering the investment.So, let me formalize this.Let H = healthcare donationEducation = 2HEnvironmental = H + 150,000Investment = ITotal: H + 2H + (H + 150,000) + I = 1,000,000Simplify:4H + 150,000 + I = 1,000,000So, 4H + I = 850,000From the investment condition:I * 1.05 ≥ 210,000So, I ≥ 210,000 / 1.05 = 200,000Therefore, I must be at least 200,000.So, plugging back into the total equation:4H + 200,000 = 850,0004H = 650,000H = 162,500Therefore, the donations are:Healthcare: 162,500Education: 325,000Environmental: 312,500Investment: 200,000So, that seems correct.But wait, the problem says "determine how much Alex donates to each cause," which are education, healthcare, and environmental conservation. So, the investment is separate from the donations? Or is it part of the donations?Wait, the problem says "donates a total of 1,000,000" and "invests a portion of the total donation amount." So, the investment is part of the 1,000,000. Therefore, the donations are the remaining amount after the investment.Therefore, in part 1, the donations are H, 2H, and H + 150,000, and the investment is I, with H + 2H + (H + 150,000) + I = 1,000,000.So, the donations are H, 2H, and H + 150,000, and the investment is I.Therefore, the answer to part 1 is the donations, which are 162,500, 325,000, and 312,500, and the investment is 200,000.But the problem is structured as two separate questions: first, determine the donations, then calculate the investment. So, perhaps part 1 is just about the donations, assuming that the investment is part of the total, but the conditions only relate to the donations.Wait, but in that case, the initial equations would have to include the investment, because the total is 1,000,000, which includes the investment.Therefore, the correct approach is to include the investment in the total, which affects the donations.So, in conclusion, the donations are:Healthcare: 162,500Education: 325,000Environmental Conservation: 312,500And the investment is 200,000.Therefore, the answers are:1. Education: 325,000; Healthcare: 162,500; Environmental Conservation: 312,500.2. The initial investment is 200,000.But wait, let me make sure that the investment is indeed part of the total donation. The problem says "invests a portion of the total donation amount," which is 1,000,000. So, yes, the investment is part of that 1,000,000. Therefore, the donations are less than 1,000,000, and the rest is invested.Therefore, the correct calculations are as above.But just to be thorough, let me consider the alternative scenario where the investment is in addition to the donations. Then, the total expenditure would be 1,000,000 + I. But the problem doesn't mention that, so it's more logical that the investment is part of the 1,000,000.Therefore, I think the correct approach is to include the investment in the total donation amount, leading to the donations being 162,500, 325,000, and 312,500, with an investment of 200,000.So, to summarize:1. Donations:- Education: 325,000- Healthcare: 162,500- Environmental Conservation: 312,5002. Investment: 200,000Yes, that seems correct.Final Answer1. Alex donates boxed{325000} to education, boxed{162500} to healthcare, and boxed{312500} to environmental conservation.2. The initial investment amount is boxed{200000}.</think>

🧠:<think>Okay, so I have this problem about a turbine blade modeled as a cantilever beam. The blade is subject to a varying load, and I need to find the deflection curve and then the maximum bending stress. Hmm, let me try to break this down step by step.First, for part 1, I need to derive the deflection curve y(x) of the blade. The blade is a cantilever beam with length L, and the load w(x) is given as w0 cos(πx/(2L)). The beam has Young's modulus E and moment of inertia I.I remember that for beams, the deflection curve can be found by solving the differential equation of the beam. The general equation is:EI (d²y/dx²) = M(x)But wait, actually, the bending moment M(x) is related to the shear force V(x), which in turn is related to the load w(x). So maybe I should start from the load and work my way up.The shear force V(x) is the integral of the load w(x) with respect to x, right? But since it's a cantilever beam, the shear force at the free end (x = L) is zero. So I can integrate w(x) from x to L to get V(x). Similarly, the bending moment M(x) is the integral of V(x) from x to L. Then, once I have M(x), I can integrate twice to get the deflection y(x), making sure to apply the boundary conditions.Let me write down the steps:1. Find the shear force V(x) by integrating the load w(x) from x to L.2. Find the bending moment M(x) by integrating V(x) from x to L.3. Use the bending moment to find the curvature (d²y/dx²) = M(x)/(EI).4. Integrate twice to find y(x), applying boundary conditions at x = 0 (fixed end) where y(0) = 0 and dy/dx(0) = 0.Wait, actually, for a cantilever beam, the boundary conditions are y(0) = 0 and dy/dx(0) = 0 because it's fixed at x = 0. At the free end x = L, the shear force and bending moment are zero, but the deflection and slope can be non-zero.So let me proceed step by step.First, the load is w(x) = w0 cos(πx/(2L)). Let me note that.Shear force V(x) is the integral of w(x) from x to L:V(x) = ∫[x to L] w(t) dt = ∫[x to L] w0 cos(πt/(2L)) dtLet me compute that integral.The integral of cos(a t) dt is (1/a) sin(a t). So here, a = π/(2L). Therefore,V(x) = w0 * [ (2L/π) sin(πt/(2L)) ] evaluated from t = x to t = L.So,V(x) = w0 * (2L/π) [ sin(πL/(2L)) - sin(πx/(2L)) ]Simplify sin(πL/(2L)) = sin(π/2) = 1.So,V(x) = w0 * (2L/π) [1 - sin(πx/(2L))]Okay, that's the shear force.Next, bending moment M(x) is the integral of V(x) from x to L:M(x) = ∫[x to L] V(t) dt = ∫[x to L] w0*(2L/π)[1 - sin(πt/(2L))] dtLet me factor out the constants:M(x) = w0*(2L/π) ∫[x to L] [1 - sin(πt/(2L))] dtCompute the integral:∫[x to L] 1 dt = L - x∫[x to L] sin(πt/(2L)) dt = [ - (2L/π) cos(πt/(2L)) ] evaluated from x to LSo,= - (2L/π) [cos(πL/(2L)) - cos(πx/(2L))]Simplify cos(πL/(2L)) = cos(π/2) = 0So,= - (2L/π) [0 - cos(πx/(2L))] = (2L/π) cos(πx/(2L))Therefore, putting it all together:M(x) = w0*(2L/π) [ (L - x) - (2L/π) cos(πx/(2L)) ]Wait, hold on. Let me double-check that.Wait, the integral of [1 - sin(πt/(2L))] dt is (L - x) + (2L/π) cos(πt/(2L)) evaluated from x to L.Wait, no, the integral of sin is negative cosine, so:∫ sin(πt/(2L)) dt = - (2L/π) cos(πt/(2L)) + CTherefore, ∫[x to L] sin(...) dt = - (2L/π)[cos(πL/(2L)) - cos(πx/(2L))] = - (2L/π)[0 - cos(πx/(2L))] = (2L/π) cos(πx/(2L))So, the integral of [1 - sin(...)] dt is (L - x) + (2L/π) cos(πx/(2L))Wait, no, hold on. The integral is:∫[x to L] 1 dt = L - x∫[x to L] sin(...) dt = (2L/π) cos(πx/(2L))Therefore, the integral of [1 - sin(...)] dt is (L - x) + (2L/π) cos(πx/(2L))Wait, no, because it's ∫[x to L] 1 dt - ∫[x to L] sin(...) dt = (L - x) - (2L/π) [cos(πx/(2L)) - cos(πL/(2L))]But cos(πL/(2L)) = 0, so it's (L - x) - (2L/π) cos(πx/(2L)).Wait, so:M(x) = w0*(2L/π) [ (L - x) - (2L/π) cos(πx/(2L)) ]Yes, that seems right.So,M(x) = (w0 * 2L/π)(L - x) - (w0 * (2L/π)^2) cos(πx/(2L))Okay, now we have M(x). Next, we can find the curvature, which is d²y/dx² = M(x)/(EI)So,d²y/dx² = [ (w0 * 2L/π)(L - x) - (w0 * (2L/π)^2) cos(πx/(2L)) ] / (EI)Let me denote this as:d²y/dx² = (w0 / EI) [ (2L/π)(L - x) - (4L²/π²) cos(πx/(2L)) ]Now, to find y(x), we need to integrate this twice.First integration gives dy/dx:dy/dx = ∫ d²y/dx² dx + C1Let me compute the integral term by term.First term: ∫ (2L/π)(L - x) dx= (2L/π) ∫ (L - x) dx= (2L/π) [ Lx - (x²)/2 ] + CSecond term: ∫ (4L²/π²) cos(πx/(2L)) dx= (4L²/π²) * (2L/π) sin(πx/(2L)) + C= (8L³/π³) sin(πx/(2L)) + CSo putting it all together:dy/dx = (w0 / EI) [ (2L/π)(Lx - x²/2) - (8L³/π³) sin(πx/(2L)) ] + C1Now, apply the boundary condition at x = 0: dy/dx(0) = 0Compute dy/dx at x = 0:= (w0 / EI) [ (2L/π)(0 - 0) - (8L³/π³) sin(0) ] + C1= 0 + C1 = 0 => C1 = 0So, dy/dx = (w0 / EI) [ (2L/π)(Lx - x²/2) - (8L³/π³) sin(πx/(2L)) ]Now, integrate dy/dx to get y(x):y(x) = ∫ dy/dx dx + C2Compute term by term.First term: ∫ (2L/π)(Lx - x²/2) dx= (2L/π) ∫ (Lx - x²/2) dx= (2L/π) [ (Lx²)/2 - (x³)/6 ] + CSecond term: ∫ (8L³/π³) sin(πx/(2L)) dx= (8L³/π³) * ( -2L/π ) cos(πx/(2L)) + C= - (16L^4 / π^4) cos(πx/(2L)) + CSo, putting it all together:y(x) = (w0 / EI) [ (2L/π)( (Lx²)/2 - x³/6 ) - (16L^4 / π^4) cos(πx/(2L)) ] + C2Simplify the first term:(2L/π)( (Lx²)/2 - x³/6 ) = (2L/π)( (3Lx² - x³)/6 ) = (2L/π)(x²(3L - x)/6 ) = (L/π)(x²(3L - x)/3 ) = (L/(3π)) x²(3L - x)Wait, let me compute it step by step:(2L/π) * (Lx²/2) = (2L/π)(Lx²/2) = (L²x²)/π(2L/π) * (-x³/6) = - (2L/π)(x³/6) = - (L x³)/(3π)So, altogether, the first part is (L²x²)/π - (L x³)/(3π)Therefore, y(x) becomes:y(x) = (w0 / EI) [ (L²x²)/π - (L x³)/(3π) - (16L^4 / π^4) cos(πx/(2L)) ] + C2Now, apply the boundary condition at x = 0: y(0) = 0Compute y(0):= (w0 / EI) [ 0 - 0 - (16L^4 / π^4) cos(0) ] + C2= (w0 / EI) [ -16L^4 / π^4 * 1 ] + C2 = 0So,- (16 w0 L^4)/(EI π^4) + C2 = 0 => C2 = (16 w0 L^4)/(EI π^4)Therefore, the deflection curve y(x) is:y(x) = (w0 / EI) [ (L²x²)/π - (L x³)/(3π) - (16L^4 / π^4) cos(πx/(2L)) ] + (16 w0 L^4)/(EI π^4)Let me factor out (w0 / EI):y(x) = (w0 / EI) [ (L²x²)/π - (L x³)/(3π) - (16L^4 / π^4) cos(πx/(2L)) + 16L^4 / π^4 ]Notice that the last two terms can be combined:- (16L^4 / π^4) cos(πx/(2L)) + 16L^4 / π^4 = (16L^4 / π^4)(1 - cos(πx/(2L)))So,y(x) = (w0 / EI) [ (L²x²)/π - (L x³)/(3π) + (16L^4 / π^4)(1 - cos(πx/(2L))) ]Alternatively, we can write it as:y(x) = (w0 / EI) [ (L²x²)/π - (L x³)/(3π) + (16L^4 / π^4)(1 - cos(πx/(2L))) ]I think that's the deflection curve.Let me check the dimensions to make sure everything makes sense.w0 has units of force per length (since it's a distributed load). L is length, E is modulus of elasticity (force per area), I is moment of inertia (length^4). So, EI has units of force * length^2.The terms inside the brackets:(L²x²)/π has units of length^4.Similarly, (L x³)/(3π) is length^4.(16L^4 / π^4)(1 - cos(...)) is length^4.So, when multiplied by w0 / EI, which is (force/length) / (force * length^2) ) = 1/length^3, multiplied by length^4 gives length, which is correct for deflection. So the units check out.I think that's part 1 done.Now, moving on to part 2: finding the maximum bending stress σ_max.The formula given is σ = M(x) c / I, where c is the distance from the neutral axis to the outermost fiber.So, σ_max will occur where M(x) is maximum, because c and I are constants for a given beam cross-section.Therefore, I need to find the maximum value of M(x) along the beam.From part 1, we have M(x) expressed as:M(x) = (w0 * 2L/π)(L - x) - (w0 * (2L/π)^2) cos(πx/(2L))Simplify:M(x) = (2 w0 L / π)(L - x) - (4 w0 L² / π²) cos(πx/(2L))To find the maximum M(x), we need to find the critical points by taking the derivative of M(x) with respect to x and setting it to zero.Compute dM/dx:dM/dx = d/dx [ (2 w0 L / π)(L - x) - (4 w0 L² / π²) cos(πx/(2L)) ]= (2 w0 L / π)(-1) - (4 w0 L² / π²)( - π/(2L) sin(πx/(2L)) )Simplify term by term:First term: -2 w0 L / πSecond term: (4 w0 L² / π²)( π/(2L) ) sin(πx/(2L)) = (4 w0 L² / π²)( π/(2L) ) sin(...) = (4 w0 L² * π ) / (2 L π² ) sin(...) = (2 w0 L ) / π sin(πx/(2L))So,dM/dx = -2 w0 L / π + (2 w0 L / π) sin(πx/(2L))Set dM/dx = 0:-2 w0 L / π + (2 w0 L / π) sin(πx/(2L)) = 0Factor out 2 w0 L / π:(2 w0 L / π)( -1 + sin(πx/(2L)) ) = 0Since 2 w0 L / π ≠ 0, we have:-1 + sin(πx/(2L)) = 0 => sin(πx/(2L)) = 1The sine function equals 1 at π/2 + 2π n, where n is integer.So,πx/(2L) = π/2 + 2π nDivide both sides by π:x/(2L) = 1/2 + 2nMultiply both sides by 2L:x = L + 4L nBut x must be between 0 and L, so n = 0.Thus, x = L.Wait, but x = L is the free end. Let me check M(x) at x = L.From M(x):M(L) = (2 w0 L / π)(L - L) - (4 w0 L² / π²) cos(πL/(2L)) = 0 - (4 w0 L² / π²) cos(π/2) = 0 - 0 = 0So, M(L) = 0. That's the free end, as expected.Hmm, so the critical point is at x = L, but that's a minimum (since M(x) is zero there). So, perhaps the maximum bending moment occurs at another point.Wait, maybe I made a mistake in solving for x.Wait, sin(πx/(2L)) = 1 => πx/(2L) = π/2 => x = LYes, that's correct. So, the only critical point in [0, L] is at x = L, but that's where M(x) is zero. Therefore, the maximum must occur at another point, perhaps at x = 0?Wait, let's check M(x) at x = 0:M(0) = (2 w0 L / π)(L - 0) - (4 w0 L² / π²) cos(0) = (2 w0 L² / π) - (4 w0 L² / π²)(1) = (2 w0 L² / π) - (4 w0 L² / π²)Hmm, let's compute that:= w0 L² [ 2/π - 4/π² ] = w0 L² [ (2π - 4)/π² ] = w0 L² (2(π - 2))/π²So, M(0) is positive.At x = L, M(L) = 0.So, since M(x) starts at M(0) positive and goes to zero at x = L, and the only critical point is at x = L where M(x) is zero, that suggests that M(x) is decreasing throughout the beam. Therefore, the maximum bending moment occurs at x = 0.Wait, but let me double-check by evaluating M(x) at some intermediate point, say x = L/2.Compute M(L/2):= (2 w0 L / π)(L - L/2) - (4 w0 L² / π²) cos(π*(L/2)/(2L)) = (2 w0 L / π)(L/2) - (4 w0 L² / π²) cos(π/4)= (w0 L² / π) - (4 w0 L² / π²)(√2/2) = (w0 L² / π) - (2√2 w0 L² / π²)Compute numerically:Let me compute coefficients:First term: 1/π ≈ 0.3183Second term: 2√2 / π² ≈ 2*1.4142 / 9.8696 ≈ 2.8284 / 9.8696 ≈ 0.2866So, M(L/2) ≈ w0 L² (0.3183 - 0.2866) ≈ w0 L² (0.0317)Which is positive but less than M(0). So, M(x) is decreasing from x=0 to x=L, with maximum at x=0.Therefore, the maximum bending moment is at x=0, which is M(0) = w0 L² [ 2/π - 4/π² ].Simplify:M_max = w0 L² (2/π - 4/π²) = w0 L² (2π - 4)/π² = w0 L² 2(π - 2)/π²So, M_max = (2 w0 L² (π - 2))/π²Therefore, the maximum bending stress σ_max is:σ_max = M_max * c / I = [ (2 w0 L² (π - 2))/π² ] * c / ISo, σ_max = (2 w0 L² c (π - 2)) / (π² I )I think that's the expression.Let me check the units again.w0 is force per length, L is length, c is length, I is length^4.So, numerator: w0 * L² * c = (force/length) * length² * length = force * lengthDenominator: I is length^4So, σ has units of force per area, which is correct.Therefore, the maximum stress occurs at x=0, the fixed end, and is given by σ_max = (2 w0 L² c (π - 2))/(π² I )I think that's the answer.Final Answer1. The deflection curve is (boxed{y(x) = frac{w_0}{EI} left( frac{L^2 x^2}{pi} - frac{L x^3}{3pi} + frac{16 L^4}{pi^4} left(1 - cosleft(frac{pi x}{2L}right)right) right)}).2. The maximum bending stress is (boxed{sigma_{text{max}} = frac{2 w_0 L^2 c (pi - 2)}{pi^2 I}}).</think>

🧠:<think>Okay, so I have this problem where a reporter is interviewing a successful novelist, and we need to figure out some math stuff about their sales and community impact. It's broken down into two sub-problems. Let me start with the first one.Sub-problem 1: Determine the coefficients ( a ), ( b ), and ( c ) for the function ( f(n) = an^2 + bn + c ). We know the sales for the first three novels: the first sold 100,000 copies, the second sold 150,000, and the third sold 210,000. So, I think I can set up a system of equations using these values.Let me write down what I know:For ( n = 1 ), ( f(1) = a(1)^2 + b(1) + c = a + b + c = 100,000 ).For ( n = 2 ), ( f(2) = a(2)^2 + b(2) + c = 4a + 2b + c = 150,000 ).For ( n = 3 ), ( f(3) = a(3)^2 + b(3) + c = 9a + 3b + c = 210,000 ).So, I have three equations:1. ( a + b + c = 100,000 ) 2. ( 4a + 2b + c = 150,000 ) 3. ( 9a + 3b + c = 210,000 )I need to solve for ( a ), ( b ), and ( c ). Let me subtract the first equation from the second to eliminate ( c ):Equation 2 - Equation 1: ( (4a + 2b + c) - (a + b + c) = 150,000 - 100,000 ) Simplify: ( 3a + b = 50,000 ) Let me call this Equation 4.Similarly, subtract Equation 2 from Equation 3:Equation 3 - Equation 2: ( (9a + 3b + c) - (4a + 2b + c) = 210,000 - 150,000 ) Simplify: ( 5a + b = 60,000 ) Let me call this Equation 5.Now, I have two equations:4. ( 3a + b = 50,000 ) 5. ( 5a + b = 60,000 )Subtract Equation 4 from Equation 5 to eliminate ( b ):( (5a + b) - (3a + b) = 60,000 - 50,000 ) Simplify: ( 2a = 10,000 ) So, ( a = 5,000 ).Now, plug ( a = 5,000 ) back into Equation 4:( 3(5,000) + b = 50,000 ) ( 15,000 + b = 50,000 ) Subtract 15,000: ( b = 35,000 ).Now, go back to Equation 1 to find ( c ):( 5,000 + 35,000 + c = 100,000 ) ( 40,000 + c = 100,000 ) Subtract 40,000: ( c = 60,000 ).So, the coefficients are ( a = 5,000 ), ( b = 35,000 ), and ( c = 60,000 ). Let me double-check these by plugging them into the original equations.For ( n = 1 ): ( 5,000(1) + 35,000(1) + 60,000 = 5,000 + 35,000 + 60,000 = 100,000 ). Correct.For ( n = 2 ): ( 5,000(4) + 35,000(2) + 60,000 = 20,000 + 70,000 + 60,000 = 150,000 ). Correct.For ( n = 3 ): ( 5,000(9) + 35,000(3) + 60,000 = 45,000 + 105,000 + 60,000 = 210,000 ). Correct.Alright, that seems solid.Sub-problem 2: The community impact is measured by an engagement index ( E(n) ), which is proportional to the square root of the number of copies sold. So, ( E(n) = k sqrt{f(n)} ), where ( k ) is the constant of proportionality.We are told that the engagement index for the third novel is 70. So, ( E(3) = 70 ). Let me write that down:( 70 = k sqrt{f(3)} ).We already know ( f(3) = 210,000 ). So,( 70 = k sqrt{210,000} ).I need to solve for ( k ). Let me compute ( sqrt{210,000} ).First, note that 210,000 is 210 * 1000, so sqrt(210,000) = sqrt(210) * sqrt(1000).Compute sqrt(210): 210 is 49*4.2857... Wait, 14^2 = 196, 15^2=225, so sqrt(210) is approximately 14.491.Compute sqrt(1000): sqrt(1000) is approximately 31.6227766.So, sqrt(210,000) ≈ 14.491 * 31.6227766 ≈ let's compute that.14 * 31.6227766 ≈ 442.7188724 0.491 * 31.6227766 ≈ approx 15.532 So total ≈ 442.7188724 + 15.532 ≈ 458.2508724.So, sqrt(210,000) ≈ 458.2509.Therefore, ( 70 = k * 458.2509 ). So, ( k = 70 / 458.2509 ≈ 0.15275 ).Let me compute that division: 70 divided by 458.2509.Well, 458.2509 * 0.15 = 68.737635 458.2509 * 0.15275 ≈ 458.2509 * 0.15 + 458.2509 * 0.00275 = 68.737635 + approx 1.258 ≈ 70. So, yes, k ≈ 0.15275.Alternatively, exact value: since 70 / sqrt(210,000) = 70 / (sqrt(210 * 1000)) = 70 / (sqrt(210) * sqrt(1000)).But maybe it's better to keep it symbolic for now.So, ( k = 70 / sqrt{210,000} ).Simplify sqrt(210,000): 210,000 = 210 * 1000 = 21 * 10 * 1000 = 21 * 10,000. Wait, no: 210,000 is 210 * 1000, which is 21 * 10 * 1000 = 21 * 10,000? Wait, 210,000 is 21 * 10,000? No, 21 * 10,000 is 210,000. Wait, 21 * 10,000 is 210,000. So, sqrt(210,000) = sqrt(21 * 10,000) = sqrt(21) * sqrt(10,000) = sqrt(21) * 100.So, sqrt(210,000) = 100 * sqrt(21). Therefore, k = 70 / (100 * sqrt(21)) = (70 / 100) / sqrt(21) = 0.7 / sqrt(21).But maybe rationalizing the denominator: 0.7 / sqrt(21) = (0.7 * sqrt(21)) / 21 = (0.7 / 21) * sqrt(21) = (1/30) * sqrt(21). Wait, 0.7 is 7/10, so 7/10 divided by 21 is 7/(10*21) = 1/30. So, yes, k = (sqrt(21))/30.Wait, let's see:( k = 70 / sqrt{210,000} = 70 / (100 sqrt{21}) = (70 / 100) / sqrt{21} = (7/10) / sqrt{21} = 7/(10 sqrt{21}) ).Multiply numerator and denominator by sqrt(21):( 7 sqrt{21} / (10 * 21) = 7 sqrt{21} / 210 = (7/210) sqrt(21) = (1/30) sqrt(21) ).So, ( k = sqrt{21}/30 ). That's a nicer exact form.So, now, we need to find the engagement index for the fifth novel, which is ( E(5) = k sqrt{f(5)} ).First, compute ( f(5) ). Since ( f(n) = 5,000n^2 + 35,000n + 60,000 ).So, ( f(5) = 5,000*(5)^2 + 35,000*5 + 60,000 ).Compute each term:5,000*(25) = 125,000 35,000*5 = 175,000 60,000 is just 60,000.Add them up: 125,000 + 175,000 = 300,000; 300,000 + 60,000 = 360,000.So, ( f(5) = 360,000 ).Therefore, ( E(5) = k sqrt{360,000} ).Compute sqrt(360,000). Well, 360,000 is 360 * 1,000, which is 36 * 10 * 1,000. Wait, sqrt(360,000) = sqrt(360 * 1,000) = sqrt(360) * sqrt(1,000).But 360,000 is 600^2, because 600*600=360,000. Wait, yes, 600^2=360,000. So, sqrt(360,000)=600.So, ( E(5) = k * 600 ).But ( k = sqrt{21}/30 ), so:( E(5) = (sqrt{21}/30) * 600 = (sqrt{21} * 600)/30 = sqrt{21} * 20 = 20 sqrt{21} ).Compute 20*sqrt(21). Since sqrt(21) is approximately 4.583666.So, 20*4.583666 ≈ 91.67332.But the question says the engagement index for the third novel is 70, so it's possible they expect an exact value or a simplified radical form.Wait, let me see: 20*sqrt(21) is exact, but maybe we can write it as 20√21. Alternatively, if they want a decimal, approximately 91.67.But let me check if I did everything correctly.Wait, let's recap:We found ( k = sqrt{21}/30 ).Then, ( E(5) = k * sqrt(f(5)) = (sqrt{21}/30) * 600 = 20 sqrt(21) ).Yes, that seems right.Alternatively, if we use the approximate value of k, which was approximately 0.15275, then:E(5) = 0.15275 * 600 ≈ 91.65, which is about the same as 20*sqrt(21) ≈ 91.65.So, whether exact or approximate, the answer is either 20√21 or approximately 91.65.But since the problem mentions that E(n) is proportional to sqrt(f(n)), and gives E(3)=70, which we used to find k, and then asks for E(5), it's likely they want an exact value, so 20√21.But let me check if I can write 20√21 in another way. 20 is 4*5, but 21 is 3*7, so it can't be simplified further. So, 20√21 is the simplest exact form.Alternatively, if they want a numerical value, maybe to the nearest whole number or something. But since 20√21 is about 91.65, which is roughly 92. But the problem doesn't specify, so perhaps better to give the exact value.Wait, let me double-check the calculation for f(5):f(n) = 5,000n² + 35,000n + 60,000.For n=5:5,000*(25) = 125,000 35,000*5 = 175,000 60,000 is 60,000.125,000 + 175,000 = 300,000; 300,000 + 60,000 = 360,000. Correct.So, sqrt(360,000)=600. Correct.So, E(5)=k*600, and k= sqrt(21)/30, so E(5)=20 sqrt(21). Correct.Alternatively, if I use the approximate value of k=0.15275, then E(5)=0.15275*600≈91.65. So, approximately 91.65.But since the problem gave E(3)=70, which is an exact number, and the function f(n) is quadratic with integer coefficients, it's possible that E(5) is also an integer or a nice fraction. But 20√21 is approximately 91.65, which is not an integer. Hmm.Wait, maybe I made a mistake in calculating k. Let me check:E(3)=70=k*sqrt(210,000). So, k=70/sqrt(210,000).But sqrt(210,000)=sqrt(210*1000)=sqrt(210)*sqrt(1000)=sqrt(210)*31.6227766.But 210=49*4.2857, but 49*4.2857 is 210. Wait, 49*4=196, 49*4.2857≈210. So, sqrt(210)=sqrt(49*4.2857)=7*sqrt(4.2857). But that's not helpful.Wait, 210,000=210*1000=21*10*1000=21*10,000. Wait, 21*10,000=210,000. So, sqrt(210,000)=sqrt(21*10,000)=sqrt(21)*sqrt(10,000)=sqrt(21)*100.Therefore, sqrt(210,000)=100 sqrt(21). So, k=70/(100 sqrt(21))=7/(10 sqrt(21))=7 sqrt(21)/(10*21)=sqrt(21)/30. Correct.So, k= sqrt(21)/30. So, E(5)=k*sqrt(f(5))=sqrt(21)/30 * 600=20 sqrt(21). So, that's correct.So, the exact value is 20√21, which is approximately 91.65. Since the problem didn't specify, but in the first part, the sales were given as whole numbers, and E(3)=70, which is a whole number, maybe they expect an exact value. So, 20√21 is the exact value.Alternatively, if we rationalize or present it differently, but I think 20√21 is the simplest exact form.So, to summarize:Sub-problem 1: a=5,000; b=35,000; c=60,000.Sub-problem 2: E(5)=20√21, which is approximately 91.65.But let me check if 20√21 is the right exact form. Since E(n)=k sqrt(f(n)), and k= sqrt(21)/30, then E(5)= (sqrt(21)/30)*600=20 sqrt(21). Yes, that's correct.Alternatively, maybe the problem expects a numerical value, but since 20√21 is exact, and 70 is exact, perhaps they want the exact form.Alternatively, maybe I can write it as 20√21 ≈91.65, but since the problem didn't specify, I think both are acceptable, but probably the exact form is better.So, final answers:Sub-problem 1: a=5,000; b=35,000; c=60,000.Sub-problem 2: E(5)=20√21.But let me check if the problem expects the answer in a specific format. It says "put your final answer within boxed{}". So, probably box each answer.But the problem is split into two sub-problems, so maybe two separate answers.But in the initial problem statement, it's one problem with two sub-problems. So, perhaps I need to present both answers.Wait, the user instruction says: "put your final answer within boxed{}". So, maybe each sub-problem's answer in a box.But I'm not sure. Maybe the assistant is supposed to answer both sub-problems in one go, but the user instruction says "put your final answer within boxed{}", so perhaps each sub-problem's answer in a box.Alternatively, maybe the assistant is supposed to write the thought process and then at the end, present both answers boxed.But in the initial problem, it's two sub-problems, so I think the answers are two separate boxed expressions.So, for Sub-problem 1: coefficients a, b, c. So, a=5,000; b=35,000; c=60,000. So, I can write them as a=5000, b=35000, c=60000.For Sub-problem 2: E(5)=20√21.So, in LaTeX, I can write:Sub-problem 1: ( a = 5000 ), ( b = 35000 ), ( c = 60000 ).Sub-problem 2: ( E(5) = 20sqrt{21} ).But to present them boxed, perhaps:For Sub-problem 1, since there are three coefficients, maybe box them together or separately. But the instruction says "put your final answer within boxed{}", so perhaps each sub-problem's answer in a box.Alternatively, if it's two separate answers, maybe two boxes.But I think the standard is to box each final answer separately.So, for Sub-problem 1, the coefficients are a=5000, b=35000, c=60000. So, I can write them as:( a = boxed{5000} ), ( b = boxed{35000} ), ( c = boxed{60000} ).But that's three boxes. Alternatively, if the answer is a set, maybe:The coefficients are ( boxed{a = 5000} ), ( boxed{b = 35000} ), and ( boxed{c = 60000} ).Alternatively, if the problem expects a single box with all three, but I think it's better to box each separately.For Sub-problem 2, the engagement index is ( boxed{20sqrt{21}} ).Alternatively, if they want a numerical value, approximately 91.65, but I think exact form is better.So, to wrap up:Sub-problem 1: a=5000, b=35000, c=60000.Sub-problem 2: E(5)=20√21.So, I think that's it.</think>