Answer:
- 6000 of Type A; 8000 of Type B; profit is 345 million
- 6000 of Type A; 4000 of Type B; profit is 45 million (your answer is correct)
Explanation:
In each case taking the partial derivative of the profit expression with respect to x and y, and setting those derivatives to zero will give two linear equations in those unknowns. Solving those equations gives the x- and y-values that will maximize profit. Putting those values into the profit expression, you can find the amount of the maximum profit.
1. p(x,y) = r(x,y) -c(x,y)
p(x,y) = 4x +6y -(x^2 -3xy +7y^2 +16x -88y -5)
∂p/∂x = 0 = 4 -2x +3y -16 = -2x +3y -12
∂p/∂y = 0 = 6 +3x -14y +88 = 3x -14y +94
Using any of several methods to solve these simultaneous equations, you find ...
(x, y) = (6, 8)
Putting x=6 and y=8 into the profit expression, you get ...
p(6,8) = 4·6 +6·8 -(6^2 -3·6·8 +7·8^2 +16·6 -88·8 -5) = 345
Sale of 6000 Type A panels and 8000 Type B panels will produce the maximum profit of $345 million.
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2. Same deal.
p(x,y) = 5x +7y -(x^2 -4xy +6y^2 +9x -17y -9)
∂p/∂x = 5 -2x +4y -9 = -2x +4y -4
∂p/∂y = 7 +4x -12y +17 = 4x -12y +24
The solution of these equations is ...
(x, y) = (6, 4)
and the maximum profit is
p(6,4) = 5*6 +7*4 -(6^2 -4*6*4 +6*4^2 +9*6 -17*4 -9) = 45
Sale of 6000 Type A panels and 4000 Type B panels will produce the maximum profit of $45 million.