MATH SOLVE

4 months ago

Q:
# 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. $

Accepted Solution

A:

Answer: How many devices should be manufactured each hour to minimize average cost?21,909What is the resulting average cost of a device?$204How does the average cost compare with the marginal cost at the optimal production level?The average cost exceeds the marginal cost in $0.18Step-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
[tex] \bf A(x)=\frac{C(x)}{x}=\frac{484000+160x+0.001x^2}{x}[/tex]
How many devices should be manufactured each hour to minimize average cost?
Taking the first derivative A'(x) with respect to x
[tex] \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}[/tex]
The points where A'(x) = 0 (critical points) are
[tex] \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[/tex]
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)
[tex] \bf A''(x)=\left(\frac{0.001x^2-480000}{x^2}\right)'=\frac{960000}{x^3}[/tex]
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):
[tex] \bf A(21,909)=\frac{484000+160(21909)+0.001(21909)^2}{21909}=\$ 204[/tex]
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.