Question

The daily cost of producing x high performance wheels for racing is given by the following​ function, where no more than 100 wheels can be produced each day. What production level will give the lowest average cost per​ wheel? What is the minimum average​ cost?C(x)=0.09x^3 - 4.5x^2 + 180x; (0,100]

Answer 1

Answer:

The production would be 25 wheels,

Lowest average cost is $ 123.75

Step-by-step explanation:

Given cost function,

$C(x) = 0.09x^3 - 4.5x^2 + 180x$

Where,

x = number of wheel,

So, the average cost per wheel,

$A(x) = \frac{C(x)}{x}=\frac{0.09x^3-4.5x^2 + 180x}{x}=0.09x^2 - 4.5x + 180$

Differentiating with respect to x,

$A'(x) = 0.18x - 4.5$

Again differentiating with respect to x,

$A''(x) = 0.18$

For maxima or minima,

$A'(x) = 0$

$0.18x - 4.5 = 0$

$0.18x = 4.5$

$\implies x = \frac{4.5}{0.18}=25$

For x = 25, A''(x) = positive,

i.e. A(x) is maximum at x = 25.

Hence, the production would be 25 wheels for the lowest average cost per​ wheel.

And, lowest average cost,

A(x) = 0.09(25)² - 4.5(25) + 180 = $ 123.75