Question

The​ cost, in​ dollars, of producing x belts is given by Upper C (x )equals 594 plus 10 x minus 0.067 x squared. Find the rate at which average cost is changing when 576 belts have been produced. When 576 belts have been​ produced, the average cost is changing at nothing dollars per belt belts per dollar for each additional belt.

Answer 1

Answer:

When 576 belts have been​ produced, the average cost is changing at 0.0652(decreasing) dollar per belt

Step-by-step explanation:

If x is the number of units of a product produced in some time interval then average cost function is given as,  

$\overline{C}=\dfrac{C\left(x\right)}{x}$

Cost function is given as,  $C\left(x\right)=594+10x-0.067x^{2}$

Substituting the value in formula for average cost function,  

$\overline{C}=\dfrac{594+10x-0.067x^{2}}{x}$

Simplifying,

$\overline{C}=\dfrac{594}{x}+\dfrac{10x}{x}-\dfrac{0.067x^{2}}{x}$

$\overline{C}=\dfrac{594}{x}+10-0.067x$

To find the rate of average cost that is, $\dfrac{d\overline{C}}{dx}$. So differentiating function $\overline{C}$ with respect to x,  

$\dfrac{d\overline{C}}{dx}=\dfrac{d}{dx}\left (\dfrac{594}{x}+10-0.067x \right )$

Applying sum rule and difference rule of derivative,

$\dfrac{d\overline{C}}{dx}=\dfrac{d}{dx}\left(\dfrac{594}{x}\right)+\dfrac{d}{dx}\left(10\right)-\dfrac{d}{dx}\left(0.067x\right)$

Applying constant multiple rule of derivative,  

$\dfrac{d\overline{C}}{dx}=594\dfrac{d}{dx}\left(\dfrac{1}{x}\right)+\dfrac{d}{dx}\left(10\right)-0.067\dfrac{d}{dx}\left(x\right)$

Since,  

$\dfrac{1}{x}=x^{-1}$

$\dfrac{d\overline{C}}{dx}=594\dfrac{d}{dx}\left(x^{-1}\right)+\dfrac{d}{dx}\left(10\right)-0.067\dfrac{d}{dx}\left(x\right)$

Applying power rule and constant rule of derivative,

$\dfrac{d\overline{C}}{dx}=594\left(-1\:x^{-1-1}\right)+0-0.067\left(1\:x^{1-1}\right)$

$\dfrac{d\overline{C}}{dx}=594\left(-1\:x^{-2}\right)-0.067\left(1\:x^{0}\right)$

$\dfrac{d\overline{C}}{dx}=594\:x^{-2}-0.067$

$\dfrac{d\overline{C}}{dx}=\dfrac{594}{x^{2}}-0.067$

Substituting the value of $x=576$,

$\dfrac{d\overline{C}}{dx}=594\left(\dfrac{1}{\left(576 \right)^{2}}\right )-0.067$

$\dfrac{d\overline{C}}{dx}=-0.0652$

Negative sign indicates that rate of average cost is decreasing.

Therefore, the average cost is changing at the rate of 0.0652(decreasing) dollar per belt