Business Applications — Question 10

Question 10

A company manufactures and sells a product. The weekly revenue and cost functions (in thousands of dollars) are: $$R(x) = 500x - 5x^2, \quad C(x) = 200x + 500$$

• Determine the profit function $P(x)$.

• Find the production level $x$ that maximizes profit.

• Compute the maximum profit.

Question 10 - Solution

(a) Profit Function: $$P(x) = R(x) - C(x) = (500x - 5x^2) - (200x + 500) = -5x^2 + 300x - 500$$

(b) Maximize Profit: $$P'(x) = -10x + 300$$ Set $P'(x) = 0$: $$-10x + 300 = 0 \quad \Rightarrow \quad x = 30$$ Check the second derivative: $$P''(x) = -10 < 0$$ Thus, profit is maximized when $x = 30$.

(c) Maximum Profit: $$P(30) = -5(30)^2 + 300(30) - 500 = -4500 + 9000 - 500 = 4000$$

$$\boxed{\text{Maximum profit occurs at } x = 30 \text{ units, } P_{\max} = 4000 \text{ thousand dollars}}$$