Questions: A firm has the marginal-demand function D'(x)=-1400x/sqrt(25-x^2), where D(x) is the number of units sold at x dollars per unit. Find the demand function given that D=11,000 when x=3 per unit. The demand function is D(x)=

Solution Steps

To find the demand function $D(x)$, we need to integrate the marginal-demand function $D'(x)$. Given that $D(3) = 11,000$, we can use this initial condition to solve for the constant of integration after performing the integration.

We are given the marginal-demand function $D'(x) = \frac{-1400x}{\sqrt{25-x^2}}$ and need to find the demand function $D(x)$. We also know that $D(3) = 11,000$.

To find the demand function $D(x)$, we need to integrate the marginal-demand function $D'(x)$.

$$D(x) = \int D'(x) \, dx = \int \frac{-1400x}{\sqrt{25-x^2}} \, dx$$

Let $u = 25 - x^2$, then $du = -2x \, dx$ or $-\frac{1}{2} du = x \, dx$.

Substitute into the integral:

$$D(x) = \int \frac{-1400x}{\sqrt{25-x^2}} \, dx = \int \frac{-1400}{\sqrt{u}} \left(-\frac{1}{2}\right) \, du$$

$$= 700 \int u^{-\frac{1}{2}} \, du$$

The integral of $u^{-\frac{1}{2}}$ is $2u^{\frac{1}{2}}$.

$$D(x) = 700 \cdot 2u^{\frac{1}{2}} + C = 1400 \sqrt{u} + C$$

Substitute back $u = 25 - x^2$:

$$D(x) = 1400 \sqrt{25 - x^2} + C$$

We know $D(3) = 11,000$.

$$11,000 = 1400 \sqrt{25 - 3^2} + C$$

$$11,000 = 1400 \sqrt{25 - 9} + C$$

$$11,000 = 1400 \sqrt{16} + C$$

$$11,000 = 1400 \times 4 + C$$

$$11,000 = 5600 + C$$

$$C = 11,000 - 5600 = 5400$$

Final Answer