Question

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

Answer 1

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))`.