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When a company produces and sells x thousand units per​ week, its total weekly profit is P thousand​ dollars, where Upper P equals StartFraction 1000 x Over 100 plus x squared EndFraction . The production level at t weeks from the present is x equals 4 plus 2 t. Find the marginal​ profit, StartFraction dY Over dx EndFraction and the time rate of change of​ profit, StartFraction dP Over dt EndFraction . How fast​ (with respect of

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Answer:

The marginal​ profit is
(1000(100-x^(2)))/((x^(2)+100)^(2)).

Explanation:

The profit function p (x) is the difference between the revenue and cost function.

The profit function is given as follows:


p (x) = (1000x)/(100+x^(2))

Determine the marginal profit function as follows:


\text{Marginal Profit}=\frac{\text{d}}{\text{dx}} p (x)


={\tfrac{\mathrm{d}}{\mathrm{d}x}\left[(1000x)/(x^2+100)\right]}}\\\\={1000\cdot{\tfrac{\mathrm{d}}{\mathrm{d}x}\left[(x)/(x^2+100)\right]}}}}\\\\


=1000\cdot\frac{\frac{\text{d}}{\text{dx}}(x)\cdot (x^(2)+100)+x\cdot \frac{\text{d}}{\text{dx}} (x^(2)+100)}{ (x^(2)+100)^(2)}\\\\=(1000(x^(2)-2x^(2)+100))/((x^(2)+100)^(2))\\\\=(1000(100-x^(2)))/((x^(2)+100)^(2))

Thus, the marginal​ profit is
(1000(100-x^(2)))/((x^(2)+100)^(2)).

User Vincent Decaux
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