Answer:
- 9000 A and 7000 B should be produced
- p = 100, q = 170
- $350,000 is the maximum profit
Step-by-step explanation:
We assume that "p" and "q" are given in dollars, so that the profit function (in thousands of dollars) can be written:
P = xp +yq -C(x, y)
Profit will be maximized when the partial derivatives of P with respect to x and y are both zero:
∂P/∂x = p +x(∂p/∂x) +y(∂q/∂x) -∂C/∂x = 0
(120 -3x +y) +x(-3) +y(1) -80 = 0
-6x +2y +40 = 0
3x -y = 20 . . . . put in standard form
and ...
∂P/∂y = x(∂p/∂y) +q +y(∂q/∂y) -∂C/∂y = 0
x(1) +(210 +x -7y) +y(-7) -(130) = 0
2x -14y +80 = 0
x -7y = -40 . . . . put in standard form
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The solution to these simultaneous equations can be found a variety of ways. Using Cramer's Rule, we have ...
x = ((-1)(-40) -(-7)(20))/(-1·1-(-7)(3)) = 180/20 = 9
y = (20(1) -(-40)(3))/20 = 140/20 = 7
9000 type A and 7000 type B earphones should be produced.
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The corresponding selling prices are ...
p = 120 -3(9) +7 = 100
q = 210 +9 -7(7) = 170
The selling prices should be ...
p = $100, q = $170
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The maximum profit is ...
P = xp +yq -C(x, y) = (9)(100) +(7)(170) -(110 +80(9) +130(7))
P = 9(100 -180) +7(170 -130) -110 = 180 +280 -110
P = 350
The maximum profit is $350,000.
