Chapter 3, Problem 25RE

25. 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

$C(x) = 4.9x^2 - 617.4x + 19,600$

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

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

To determine

To calculate:

a. How many cubes should be manufactured to minimise the marginal cost.

b. At this level of production what is the marginal cost.

Answer to Problem 25RE

a. 63 cubes should be manufactured to minimise the marginal cost.

b. The minimum marginal cost is $\$120.4$.

Explanation of Solution

Given:

The marginal cost $C$ of manufacturing $x$ golf clubs is given by the equation

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

Formula used:

For a quadratic function $f(x)=a{x}^2+bx+c$, the vertex $\left(\frac{-b}{2a},f\left(\frac{-b}{2a}\right)\right)$ is the maximum point if $a$ is negative and the vertex is the minimum point if $a$ is positive.

Calculation:

From the given equation, we can see that the marginal cost equation is a quadratic equation with $a=4.9>0$ positive.

Therefore the vertex is the minimum point.

Thus, we have

$x=\frac{-b}{2a}=\frac{-(-617.4)}{2(4.9)}=63$

a. Thus, 63 cubes should be manufactured to minimise the marginal cost.

Now, we have to find the minimum marginal cost.

Thus, we have

$C(63)=4.9{(63)}^2-617.4(63)+19600=120.4$

b. The minimum marginal cost is $\$120.4$.