Certainly! Let's find the consumer's surplus for the given price-demand equation [tex]\( p = D(x) = \frac{12,000 - 40x}{400 - x} \)[/tex] at the price level [tex]\( \bar{p} = \$20 \)[/tex].
### Step-by-Step Solution
1. Determine the quantity demanded [tex]\( x \)[/tex] at the price [tex]\( \bar{p} \)[/tex]:
The relationship between price [tex]\( p \)[/tex] and quantity demanded [tex]\( x \)[/tex] is given by:
[tex]\[
p = \frac{12,000 - 40x}{400 - x}
\][/tex]
To find the quantity demanded [tex]\( \bar{x} \)[/tex] at [tex]\( \bar{p} = 20 \)[/tex], solve the equation:
[tex]\[
20 = \frac{12,000 - 40x}{400 - x}
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
20 (400 - x) = 12,000 - 40x
\][/tex]
[tex]\[
8,000 - 20x = 12,000 - 40x
\][/tex]
[tex]\[
20x = 4,000
\][/tex]
[tex]\[
x = 66.6667
\][/tex]
2. Set up the integral to find the consumer's surplus:
Consumer's surplus is the area between the demand curve and the price level [tex]\( \bar{p} \)[/tex] from 0 to [tex]\( \bar{x} \)[/tex]. Mathematically, it is represented as:
[tex]\[
\text{Consumer's Surplus} = \int_0^{\bar{x}} \left( D(x) - \bar{p} \right) dx
\][/tex]
3. Integrate the function [tex]\( D(x) - \bar{p} \)[/tex]:
The demand function [tex]\( D(x) = \frac{12,000 - 40x}{400 - x} \)[/tex]. So, we need to compute:
[tex]\[
\int_0^{66.6667} \left( \frac{12,000 - 40x}{400 - x} - 20 \right) dx
\][/tex]
4. Simplify the integrand:
[tex]\[
\frac{12,000 - 40x}{400 - x} - 20 = \frac{12,000 - 40x - 20(400 - x)}{400 - x} = \frac{12,000 - 40x - 8,000 + 20x}{400 - x} = \frac{4,000 - 20x}{400 - x}
\][/tex]
5. Perform the integration:
Use a table of integrals (or computational tools, though not mentioned explicitly):
[tex]\[
\int \frac{4,000 - 20x}{400 - x} \, dx = \text{Some antiderivative function evaluated from 0 to 66.6667}
\][/tex]
After evaluating the integral from [tex]\( x = 0 \)[/tex] to [tex]\( x = 66.6667 \)[/tex], we get:
[tex]\[
\text{Consumer's Surplus} \approx 604 \text{ dollars}
\][/tex]
6. Round the consumer's surplus to the nearest dollar:
The resulting consumer surplus is approximately \[tex]$604.
### Final Answer
The consumers' surplus is \$[/tex]604.
