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.
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.
![](/static/compass_v2/shared-icons/check-mark.png)
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 3 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)