Question

Assume that it costs a company approximately C(x) = 484,000 + 160x + 0.001x2 dollars to manufacture x units of a device in an hour at one of their manufacturing centers. How many devices should be manufactured each hour to minimize average cost? units What is the resulting average cost of a device? $ How does the average cost compare with the marginal cost at the optimal production level? Find how much they differ. $

Answer 1

Answer:

How many devices should be manufactured each hour to minimize average cost?

21,909

What is the resulting average cost of a device?

$204

How does the average cost compare with the marginal cost at the optimal production level?

The average cost exceeds the marginal cost in  $0.18

Step-by-step explanation:

The average cost A(x) equals the total cost C(x) divided by the number x of units produced in a given period. So

$\bf A(x)=\frac{C(x)}{x}=\frac{484000+160x+0.001x^2}{x}$

How many devices should be manufactured each hour to minimize average cost?

Taking the first derivative A'(x) with respect to x

$\bf A'(x)=\left(\frac{484000+160x+0.001x^2}{x}\right)'=\\=\frac{(484000+160x+0.001x^2)'x-(484000+160x+0.001x^2)x'}{x^2}=\\=\frac{(160+0.002x)x-(484000+160x+0.001x^2)}{x^2}=\frac{0.001x^2-480000}{x^2}$

The points where A'(x) = 0 (critical points) are

$\bf A'(x)=0\Rightarrow\frac{0.001x^2-480000}{x^2}=0\Rightarrow 0.001x^2=480000\Rightarrow\\\Rightarrow x^2=\frac{480000}{0.001}\Rightarrow x^2=480,000,000\Rightarrow x=\pm\sqrt{480,000,000}\Rightarrow\\\Rightarrow x=\pm 21908.9023$

So, x=21,908.9023 and x = -21,9023 are the two critical points.

To find out which one is a minimum we take the second derivative A''(x)

$\bf A''(x)=\left(\frac{0.001x^2-480000}{x^2}\right)'=\frac{960000}{x^3}$

and A''( 21,908.9023) > 0 , so x = 21,908.9023 is a minimum.

Given that x must be an integer  

x = 21,909

is the number of units that minimizes the average cost.

What is the resulting average cost of a device?

It would be A(21,909):

$\bf A(21,909)=\frac{484000+160(21909)+0.001(21909)^2}{21909}=\ 204$

How does the average cost compare with the marginal cost at the optimal production level? Find how much they differ.

The marginal cost is  

C'(x) = 160 + 0.002x

hence

C'(21,909) = 160 + 0.002(21909) = $203.82

and the average cost exceeds the marginal cost in

204 - 203.82 = $0.18

at the optimal  production level.