Business Applications — Question 2

Question 2

A company manufactures and sells $x$ units of a product per week. The revenue (in thousands of dollars) and cost (in thousands of dollars) functions are given by: $$R(x) = 50x - 0.5x^2, \quad C(x) = 20x + 100$$

• Find the profit function $P(x)$.

• Determine the number of units $x$ that maximizes the profit.

• Compute the maximum profit.

Question 2 - Solution

(a) Profit Function:

$$P(x) = R(x) - C(x) = (50x - 0.5x^2) - (20x + 100) = -0.5x^2 + 30x - 100$$

(b) Maximize Profit:

$$P'(x) = -x + 30$$

Set $P'(x) = 0$: $$-x + 30 = 0 \quad \Rightarrow \quad x = 30$$

Second derivative: $$P''(x) = -1 < 0$$ Hence, profit is maximized at $x = 30$.

(c) Maximum Profit:

$$P(30) = -0.5(30)^2 + 30(30) - 100 = -450 + 900 - 100 = 350$$

$$\boxed{ \text{Maximum profit occurs at } x = 30 \text{ units, } P_{\max} = 350 \text{ thousand dollars} }$$