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The​ cost, in​ dollars, of producing x belts is given by Upper C (x )equals 970 plus 12 x minus 0.076 x squared. The​ revenue, in​ dollars, of producing and selling x belts is given by Upper R (x )equals 43 x Superscript three fourths . Find the rate at which average profit is changing when 484 belts have been produced and sold.

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Answer:

$68.44 for each additional belt

Step-by-step explanation:

The cost and revenue functions are:

[tex]C(x) = 970+12x - 0.076x^2\\R(x) =43x^{\frac{3}{4}}[/tex]

The profit function, P(x), is given by subtracting the cost function from the revenue function.

[tex]P(x) = R(x) - C(x)\\P(x)=43x^{\frac{3}{4}}+0.076x^2-12x-970\\[/tex]

The derivate of the profit function gives us the rate at which profit changes:

[tex]\frac{dP(x)}{dx} =P'(x)=32.25x^{\frac{-1}{4}}+0.152x-12\\[/tex]

For x = 484, the rate of change is:

[tex]P'(484)=32.25*484^{\frac{-1}{4}}+0.152*484-12\\P'(484) = \$68.44/belt[/tex]

Profit is increasing by $68.44 for each additional belt

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