The maximum value occurs at x = 208.5 when the function is P(q) = 417q - q² - 200
Explanation:
Given:
R(q) = 425q - q²
C(q) = 200 + 8q
Profit function, P(q) = ?
The profit function, P(q) is given by the difference between the revenue and cost function.
P(q) = R(q) - C(q)
P(q) = 425q - q² - 200 - 8q
P(q) = 417q - q² - 200
The above profit function is a downward opening parabola. Its maximum value occurs at:
[tex]x = -\frac{b}{2a} = \frac{417}{2} = 208.5[/tex]
Therefore, maximum value occurs at x = 208.5 when the function is P(q) = 417q - q² - 200
The maximum profit is $43272 which is made from selling 209 units.
Revenue is the amount of money made from selling a particular number of items while cost is the total money spent to produce a particular number of items.
Profit is the difference between revenue and cost. It is given by:
Profit = Revenue - Cost
Given that R(q) = 425q − q², C(q) = 200 + 8q
Profit (P) = 425q − q² - (200 + 8q) = 417q - q² - 200
P = 417q - q² - 200
The maximum profit is at dP/dq = 0:
dP/dq = 417 - 2q
0 = 417 - 2q
2q = 417
q = 208.5 ≅ 209
P(209) = 417(209) - (209)² - 200 = 43272
Therefore the maximum profit is $43272 which is made from selling 209 units.
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