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?

Short answer

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

b) the minimum marginal cost is $2000

Step-by-step solution

Part a) Step 1: Given information

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

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

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

Conclusion

a)The minimum number of cell phones produced is 20

b) The minimum marginal cost is $2000