Casper's utility function is 3x + y, where x is his consumption of cocoa and y is his consumption of cheese.If the total cost of x units of cocoa is x², the price of cheese is $8, and Casper's income is $174, how many units of cocoa will he consume? A)9 B)11 C)12 D)23 E)24

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We need to maximize Casper's utility function subject to his budget constraint. First, we need to calculate his budget constraint, which is: px * x + py * y = I where px is the price of cocoa, x is the units of cocoa, py is the price of cheese ($8) and y is the units of cheese, and I is Casper's income ($174). From the information given, we know that px = x², which means that: px * x = x(x²) = x^2² So our budget constraint becomes: x^2² + 8y = 174 We can solve for y in terms of x: y = (174 - x^2²) / 8 Now we can substitute this expression for y into Casper's utility function: U(x) = 3x F1F1F1S1F1F1F10 y U(x) = 3x F1F1F1S1F1F1F10 [(174 - x^2²) / 8] We want to find the value of x that maximizes U(x). To do this, we take the derivative of U(x) with respect to x and set it equal to zero: dU/dx = 3 F1F1F1S1F1F1F10 [(174 - x^2²) / 8] - 6x² = 0 Simplifying this expression: 3 F1F1F1S1F1F1F10 [(174 - x^2²) / 8] = 6x² 174 - x^2² = 16x² x^2² + 16x² - 174 = 0 We can solve this quadratic equation using the quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / 2a where a = 1, b = 16², and c = -174² Plugging in the values: x = (-16² ± sqrt((16²)^2 - 4(1)(-174²))) / 2(1) Using a calculator, we get two solutions: x ≈ 11.09 or x ≈ -27.09 We ignore the negative solution, so Casper will consume approximately 11 units of cocoa. Therefore, the best choice is C.

Casper's Utility Function Is 3x + Y, Where X Is