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 pharmaceutical 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.