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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
[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.
The figures related to the average cost of the devices to be manufactured are:
- Approximately 22,000 items are needed to be manufactured to be manufactured approximately to minimize the manufacturing cost.
- The average cost at x = 22,000 is $204
- The average cost at this production rate is > marginal cost
- The difference between them is $22
How to obtain the maximum value of a function?
To find the maximum of a continuous and twice differentiable function f(x), we can firstly differentiate it with respect to x and equating it to 0 will give us critical points.
Putting those values of x in the second rate of function, if results in negative output, then at that point, there is maxima. If the output is positive then its minima and if its 0, then we will have to find the third derivative (if it exists) and so on.
How to find the marginal cost?
The derivative of the cost function with respect to the number of units manufactured is called marginal cost.
How to find the average cost?
The total cost for 'x' items divided by 'x' gives the average cost for 'x' items.
For the considered case, the average cost of producing 'x' units to the considered company is given by:
[tex]C(x) = 484000 + 160x + 0.001x^2[/tex]
The marginal cost for 'x' units/hour to be manufactured would be:
[tex]\dfrac{dC(x)}{dx} = 160 + 0.002x[/tex]
The average cost function for manufacturing 'x' items is given by:
[tex]f(x) = \dfrac{C(x)}{x} = \dfrac{484000}{x} + 160 + 0.001x[/tex]
Finding its first derivative with respect to 'x', we get:
[tex]f'(x) = 0.001 - \dfrac{484000}{x^2}[/tex]
Its second derivative comes out as:
[tex]f''(x) = \dfrac{484000}{x^3}[/tex]
This is going to be positive for all positive values of 'x' (x has to be positive as it represents units of devices manufactured/hour)
Therefore, since second rate > 0, all critical values will represent minima point.
Equating first derivative to 0 to find the critical points, we get:
[tex]f'(x) = 0.001 - \dfrac{484000}{x^2} = 0\\\\\x = \sqrt{484000 \times 1000} = 1000\sqrt{484} = 22000[/tex]
Thus, 22000 items are needed to be manufactured to be manufactured to minimize the manufacturing cost.
The average cost at this value of 'x' is:
[tex]f(x) = \dfrac{484000}{x} + 160 + 0.001x\\\\f(22000) = \dfrac{484000}{22000} + 160 + 0.001 \times 22000 = 204 \text{ (in dollars) }[/tex]
At this value of x, the marginal cost evaluates to:
[tex]\dfrac{dC(x)}{dx} = 160 + 0.002x = 160 + 22000\times 0.001 = 182\text{ (in dollars) }[/tex]
Thus, the average cost at this production rate is > marginal cost
The difference between them is $22
Thus, the figures related to the average cost of the devices to be manufactured are:
- Approximately 22,000 items are needed to be manufactured to be manufactured approximately to minimize the manufacturing cost.
- The average cost at x = 22,000 is $204
- The average cost at this production rate is > marginal cost
- The difference between them is $22
Learn more about marginal cost and average cost here:
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