Question

A fabrics company finds that the cost and​ revenue, in​ dollars, of producing x jackets is given by ​C(x)equals=940 plus 13 StartRoot x EndRoot940+13x and ​R(x)equals=83 StartRoot x EndRoot83x​, respectively. Determine the rate at which the fabric​ company's average profit per jacket is changing when 500500 jackets have been produced and sold.

Answer 1

Given that a fabrics company finds that the cost and revenue, in dollars, of producing x jackets is given by
$C(x)=940+13 \sqrt{x}$
and
$R(x)=83 \sqrt{x}$
respectively.

The profit of the company from producing x jackets is given by P(x) = R(x) - C(x) = $83 \sqrt{x}-\left(940+13 \sqrt{x}\right)=70 \sqrt{x} -940$

The rate at which the fabric company's average profit per jacket is changing is given by
$\frac{dP(x)}{dx} = \frac{d}{dx} (70 \sqrt{x} -940)= \frac{35}{ \sqrt{x} }$

Therefore, the the rate at which the fabric​ company's average profit per jacket is changing when 500 jackets have been produced and sold is given by
$\frac{35}{ \sqrt{x}}= \frac{35}{ \sqrt{500} } =\ 1.57$