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

Answer:

The marginal​ profit is $\frac{1000(100-x^{2})}{(x^{2}+100)^{2}}$.

Step-by-step explanation:

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

The profit function is given as follows:

$p (x) = \frac{1000x}{100+x^{2}}$

Determine the marginal profit function as follows:

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

                        $={\tfrac{\mathrm{d}}{\mathrm{d}x}\left[\dfrac{1000x}{x^2+100}\right]}}\\\\={1000\cdot{\tfrac{\mathrm{d}}{\mathrm{d}x}\left[\dfrac{x}{x^2+100}\right]}}}}$

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

Thus, the marginal​ profit is $\frac{1000(100-x^{2})}{(x^{2}+100)^{2}}$.