Q. 90 - Precalculus Enhanced with Graphing Utilities

Question

The marginal cost C (in dollars) of manufacturing x cell phones (in thousands) is given by

$C\left(x\right) = 5x^2 - 200x + 4000$

(a) How many cell phones should be manufactured to minimize the marginal cost?

(b) What is the minimum marginal cost?

Step-by-Step Solution

Answer

a) The number of cell phones produced should be is 20

b) the minimum marginal cost is $2000

Part a) Step 1: Given information

The given quadratic equation $C\left(x\right)=5x^2-200x+4000$

Step 2: Find the vertex

As the coefficient of $x^2$is positive the parabola opens upward. the vertex is the point of minimum

The vertex can be given as

$$\begin{gathered} x=-\frac{b}{2a} \\ x=-\frac{-200}{10} \\ x=20 \end{gathered}$$

The number of cell phones produced is 20

Part b) Step 1: To find the minimum marginal cost

Put 20 for $x$ in the equation 

$$\begin{gathered} f\left(x\right)=5x^2-200x+4000 \\ f\left(20\right)=2000-4000+4000 \\ f\left(20\right)=2000 \end{gathered}$$

Step 4: Conclusion

a)The minimum number of cell phones produced is 20

b) The minimum marginal cost is $2000