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.
Given that a fabrics company finds that the cost and revenue, in dollars, of producing x jackets is given by [tex]C(x)=940+13 \sqrt{x} [/tex] and [tex]R(x)=83 \sqrt{x} [/tex] respectively.
The profit of the company from producing x jackets is given by P(x) = R(x) - C(x) = [tex]83 \sqrt{x}-\left(940+13 \sqrt{x}\right)=70 \sqrt{x} -940[/tex]
The rate at which the fabric company's average profit per jacket is changing is given by [tex] \frac{dP(x)}{dx} = \frac{d}{dx} (70 \sqrt{x} -940)= \frac{35}{ \sqrt{x} } [/tex]
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 [tex]\frac{35}{ \sqrt{x}}= \frac{35}{ \sqrt{500} } =\$1.57[/tex]
