To find the utility-maximizing combination of goods x and y for this consumer, we can use the concept of consumer optimization subject to a budget constraint. The consumer's goal is to maximize utility (U) given the prices of the goods (px and py) and their income (I). The budget constraint can be written as: px * x py * y = I In this case, px = 4, py = 10, and I = 1200. Therefore, the budget constraint becomes: 4x 10y = 1200 Now, we can set up the consumer's optimization problem, which is to maximize the utility function U = 4x^2 9y^2 subject to the budget constraint: Maximize U = 4x^2 9y^2 Subject to: 4x 10y = 1200 We can use the method of Lagrange multipliers to solve this problem. The Lagrangian function is given by: L(x, y, ) = 4x^2 9y^2 - (4x 10y - 1200) Now, we can find the first-order partial derivatives with respect to x, y, and and set them equal to zero to find the critical points: L/x = 8x - 4 = 0 L/y = 18y - 10 = 0 L/ = -(4x 10y - 1200) = 0 Solving this system