Business Applications — Question 5

Question 5

A company produces and sells $x$ units of a product per week. The weekly revenue and cost functions (in thousands of dollars) are: $$R(x) = 200x - 2x^2, \quad C(x) = 50x + 300$$

• Find the profit function $P(x)$.

• Determine the number of units that maximizes the profit.

• Compute the maximum profit.

Question 5 - Solution

$$P(x)=R(x)-C(x)=-2x^2+150x-300.$$

In the continuous model, $P'(x)=-4x+150=0$ gives $x=37.5$ and $P''(x)=-4<0$.

The continuous maximum is $2512.5$ thousand dollars.

For whole units, compare $x=37$ and $x=38$, the integers nearest the vertex:

$$P(37)=P(38)=2512.$$

Thus the feasible maximum is

$$\boxed{37\text{ or }38\text{ units},\qquad2512\text{ thousand dollars}.}$$