The revenue from selling q items is R ( q ) = 425 q − q 2 , and the total cost is C ( q ) = 200 + 8 q . Write a function that gives the total profit earned, and find the quantity which maximizes the profit.

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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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:

Therefore, maximum value occurs at x = 208.5 when the function is P(q) = 417q - q² - 200

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