Question

Suppose that c (x )equals 4 x cubed minus 16 x squared plus 12 comma 000 x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items.

Answer 1

Answer:

Therefore the number of product is 11.42 to minimize the average cost .

Step-by-step explanation:

Given that,

$C(x)=4x^3-16x^2+12,000x$

where C(x) is cost of manufacturing of x items.

Differentiating with respect to x

$C'(x)=12x^2-32x+12,000$

Again differentiating with respect to x

$C''(x)=24x-32$

To find the minimum cost, we set C'(x)=0

$\therefore 12x^2-32x-1200=0$

$\Rightarrow 4(3x^2-8x-300)=0$

$\Rightarrow 3x^2-8x-300=0$     [ since 4≠0]

Applying quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, here a= 3, b= -8, c=-300

$\therefore x=\frac {-(-8)\pm\sqrt{(-8)^2-4.3.(-300)}}{2.3}$

      =11.42, -8.76

The number of item is negative, it can't make sense.

∴x=11.42

Now

$C''|_{x=11.42}=24(11.42)-32=242.08>0$    

Therefore when x= 11.42, the cost of manufacturing will minimum.

Therefore the number of product is 11.42 to minimize the average cost .