Q. 25 - Precalculus Enhanced with Graphing Utilities

Question

Minimizing Marginal Cost Callaway Golf Company has determined that the marginal cost C of manufacturing x Big Bertha golf clubs may be expressed by the quadratic function

$C\left(x\right)=4.9x^2-617.4x+19600$

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

(b) At this level of production, what is the marginal cost? 

Step-by-Step Solution

Answer

Part (a). To minimize the marginal cost 63 clubs should be manufactured.

Part (b). The marginal cost is 151.9 dollars. 

Step 1. Given information.

The given quadratic function 

$C\left(x\right)=4.9x^2-617.4x+19600$.

Part (a) Step 1. Find how many clubs should be manufactured to minimize the marginal cost.

The marginal cost is expressed by the quadratic function:

$C\left(x\right)=4.9x^2-617.4x+19600$

Here the parabola is open up, because $a>0$, which means the function has minimum in the vertex.

The vertex has coordinates:$\left(-\frac{b}{2a}, C\left(-\frac{b}{2a}\right)\right)$.

So, $-\frac{b}{2a}=-\frac{617.4}{2·(4.9)}=63$

To minimize the marginal cost 63 clubs should be manufactured.

Part (b) Step 1. At this level of production, what is the marginal cost?

The goal of this task is to find the marginal cost if $x=63$.

The marginal cost is expressed by the quadratic function:

$$\begin{gathered} C\left(x\right)=4.9x^2-617.4x+19600 \\ C\left(63\right)=4.9{\left(63\right)}^2-617.4\left(63\right)+19600 \\ C\left(63\right)=4.9·\left(3969\right)-38896.2+19600 \\ C\left(63\right)=151.9 \end{gathered}$$

So, the marginal cost is 151.9 dollars.