A company makes two products, A and B, from the same materials and needs to deci

Question

A company makes two products, A and B, from the same materials and needs to decide how to allocate limited resources to the production of A and B. If it produces x units of A and y units of B, then x and y satisfy the equation

which is called a production possibilities curve and is shown in the figure.

Each point (x, y) on this curve represents a production schedule specifying that x units of A and y units of B should be made. If each unit of A yields a profit of $6 and each unit of B yields a profit of $10, then the profit function is

Determine the production schedule that maximizes profit.

Explanation / Answer

To maximize = 6x+10y

Constraint = 36x^2 + 25y^2 = 90000

Using lagrange's multipliers we get

6 = 72kx ==> x = 1/12k

10 = 50ky ==>y = 1/5k

36x^2 + 25y^2 = 90000

36/144k^2 +25/25k^2 = 90000

90000k^2 = 1+1/4=5/4

k^2 = 1/72000

k = 1/120sqrt(5)

x = 1/12k = 10sqrt(5)

y = 1/5k = 24sqrt(5)

(x,y) = (10sqrt(5),24sqrt(5))

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