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)=

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)=

Transcript text: A firm has the marginal-demand function $D^{\prime}(x)=\frac{-1400 x}{\sqrt{25-x^{2}}}$, where $D(x)$ is the number of units sold at $x$ dollars per unit. Find the demand function given that $\mathrm{D}=11,000$ when $\mathrm{x}=\$ 3$ per unit. The demand function is $D(x)=$ $\square$

Solution

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.

Step 1: Understand the Problem

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 $.

Step 2: Integrate the Marginal-Demand Function

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

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

Step 3: Use Substitution for Integration

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

Substitute into the integral:

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

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

Step 4: Integrate

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 $: