In economics, the usefulness or utility of amounts x and y of two capital goods G1 and G2 is sometimes measured by a function U(x, y). For example, G1 and G2 might be two chemicals a phar-maceutical company needs to have on hand and U(x, y) the gain from manufacturing a product whose synthesis requires different amounts of the chemicals depending on the process used. If G1 costs a dollars per kilogram, G2 costs b dollars per kilogram, and the total amount allocated for the purchase of G1 and G2 together is c dollars, then the company’s managers want to maximize U(x, y) given that ax + by = c. Thus, they need to solve a typical Lagrange multiplier problem.Suppose thatU(x, y) = xy + 2xand that the equation ax + by = c simplifies to 2x + y = 30.Find the maximum value of U and the corresponding values of x and y subject to this latter constraint.

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Asked Mar 5, 2020
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In economics, the usefulness or utility of amounts x and y of two capital goods G1 and G2 is sometimes measured by a function U(x, y). For example, G1 and G2 might be two chemicals a phar-maceutical company needs to have on hand and U(x, y) the gain from manufacturing a product whose synthesis requires different amounts of the chemicals depending on the process used. If G1 costs a dollars per kilogram, G2 costs b dollars per kilogram, and the total amount allocated for the purchase of G1 and G2 together is c dollars, then the company’s managers want to maximize U(x, y) given that ax + by = c. Thus, they need to solve a typical Lagrange multiplier problem.

Suppose that

U(x, y) = xy + 2x

and that the equation ax + by = c simplifies to 2x + y = 30.

Find the maximum value of U and the corresponding values of x and y subject to this latter constraint.

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