The total cost (in hundreds of dollars) to produce x units of a product is c(x) = (3x-2) / (8x+1), find the average cost for eac
h of the following production.a) 50 unitsb) x unitsc) find the marginal average cost function.
1 answer:
Answer:
a) ![\frac{74}{10025}](https://tex.z-dn.net/?f=%5Cfrac%7B74%7D%7B10025%7D)
b) ![\frac{3x-2}{x(8x+1)}](https://tex.z-dn.net/?f=%5Cfrac%7B3x-2%7D%7Bx%288x%2B1%29%7D)
c) ![\frac{-24x^2+32x-2}{(8x^2+x)^2}](https://tex.z-dn.net/?f=%5Cfrac%7B-24x%5E2%2B32x-2%7D%7B%288x%5E2%2Bx%29%5E2%7D)
Step-by-step explanation:
For total cost function
, average cost is given by
i.e., total cost divided by number of units produced.
Marginal average cost function refers to derivative of the average cost function i.e., ![\left ( \frac{c(x)}{x} \right )'](https://tex.z-dn.net/?f=%5Cleft%20%28%20%5Cfrac%7Bc%28x%29%7D%7Bx%7D%20%5Cright%20%29%27)
Given:![c(x)=\frac{3x-2}{8x+1}](https://tex.z-dn.net/?f=c%28x%29%3D%5Cfrac%7B3x-2%7D%7B8x%2B1%7D)
Average cost = ![\frac{c(x)}{x}=\frac{3x-2}{x(8x+1)}](https://tex.z-dn.net/?f=%5Cfrac%7Bc%28x%29%7D%7Bx%7D%3D%5Cfrac%7B3x-2%7D%7Bx%288x%2B1%29%7D)
a)
At x = 50 units,
![\frac{c(50)}{50}=\frac{150-2}{50(400+1)}=\frac{148}{50(401)}=\frac{74}{10025}](https://tex.z-dn.net/?f=%5Cfrac%7Bc%2850%29%7D%7B50%7D%3D%5Cfrac%7B150-2%7D%7B50%28400%2B1%29%7D%3D%5Cfrac%7B148%7D%7B50%28401%29%7D%3D%5Cfrac%7B74%7D%7B10025%7D)
b)
Average cost = ![\frac{c(x)}{x}=\frac{3x-2}{x(8x+1)}](https://tex.z-dn.net/?f=%5Cfrac%7Bc%28x%29%7D%7Bx%7D%3D%5Cfrac%7B3x-2%7D%7Bx%288x%2B1%29%7D)
c)
Marginal average cost:
Differentiate average cost with respect to ![x](https://tex.z-dn.net/?f=x)
Take ![f=3x-2\,,\,g=8x^2+x](https://tex.z-dn.net/?f=f%3D3x-2%5C%2C%2C%5C%2Cg%3D8x%5E2%2Bx)
using quotient rule, ![\left ( \frac{f}{g} \right )'=\frac{f'g-fg'}{g^2}](https://tex.z-dn.net/?f=%5Cleft%20%28%20%5Cfrac%7Bf%7D%7Bg%7D%20%5Cright%20%29%27%3D%5Cfrac%7Bf%27g-fg%27%7D%7Bg%5E2%7D)
Therefore,
![\left ( \frac{c(x)}{x} \right )'=\left ( \frac{3x-2}{8x^2+x} \right )'\\=\left ( \frac{3(8x^2+x)-(16x+1)(3x-2)}{(8x^2+x)^2} \right )\\=\frac{24x^2+3x-48x^2-3x+32x+2}{(8x^2+x)^2}\\=\frac{-24x^2+32x-2}{(8x^2+x)^2}](https://tex.z-dn.net/?f=%5Cleft%20%28%20%5Cfrac%7Bc%28x%29%7D%7Bx%7D%20%5Cright%20%29%27%3D%5Cleft%20%28%20%5Cfrac%7B3x-2%7D%7B8x%5E2%2Bx%7D%20%5Cright%20%29%27%5C%5C%3D%5Cleft%20%28%20%5Cfrac%7B3%288x%5E2%2Bx%29-%2816x%2B1%29%283x-2%29%7D%7B%288x%5E2%2Bx%29%5E2%7D%20%5Cright%20%29%5C%5C%3D%5Cfrac%7B24x%5E2%2B3x-48x%5E2-3x%2B32x%2B2%7D%7B%288x%5E2%2Bx%29%5E2%7D%5C%5C%3D%5Cfrac%7B-24x%5E2%2B32x-2%7D%7B%288x%5E2%2Bx%29%5E2%7D)
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