9514 1404 393
Answer:
a) R = 40x -0.1x²
b) loss: x < 90 or x > 260; profit: 90 < x < 260; break-even: x = 90 or 260
c) P(x) = -0.1x² +35x -2340
d) P'(75) = 20, decrease in loss by selling one more item
Step-by-step explanation:
a) Revenue is the product of number sold (x) and the price at which they are sold (p(x)).
R(x) = x·p(x) = x(40 -0.1x)
R(x) = -0.1x² +40x
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b) The break-even points are the values of x where revenue is equal to cost.
R(x) = C(x)
-0.1x² +40x = 5x +2340
0.1x² -35x +2340 = 0 . . . . . . subtract the left-side expression
x² -350x +23400 = 0 . . . . . . multiply by 10
(x -90)(x -260) = 0 . . . . . . . . . factor
x = 90, x = 260 . . . . . . . . . values of x to break even
The company will break even with sales of 90 or 260 units.
The company will profit with sales between 90 and 260 units; it will have a loss for sales less than 90 or greater than 260 units.
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c) Profit is the difference between revenue and cost.
P(x) = R(x) -C(x)
P(x) = -0.1x² +35x -2340 . . . . . . . the opposite of C -R in part (b)
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d) Marginal profit is the derivative of the profit function.
P'(x) = -0.2x +35
Then for 75 units, the marginal profit is ...
P'(75) = -0.2×75 +35 = -15 +35 = 20
The marginal profit at x=75 is 20. The increase in profit from sale of 1 more unit is $20 when the number of units sold is 75.
