Certainly! Let's work through the problems step by step.
### Finding the Equilibrium Quantity
1. Setting up the Equilibrium Condition:
To find the equilibrium quantity, we set the demand function equal to the supply function. This is because at equilibrium, the quantity of items demanded is equal to the quantity of items supplied.
[tex]\[
d(x) = s(x)
\][/tex]
2. Substitute the given functions:
[tex]\[
200 - 0.6x = 0.2x
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Combine like terms:
[tex]\[
200 = 0.8x
\][/tex]
Divide both sides by 0.8 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{200}{0.8} = 250
\][/tex]
So, the equilibrium quantity is:
[tex]\[
x = 250
\][/tex]
### Finding the Consumer Surplus at the Equilibrium Quantity
1. Substitute the equilibrium quantity into the demand function to find the equilibrium price:
[tex]\[
\text{Equilibrium Price} = d(250) = 200 - 0.6 \times 250
\][/tex]
Calculate the price:
[tex]\[
200 - 150 = 50
\][/tex]
2. Consumer Surplus:
The consumer surplus is the area between the demand curve and the equilibrium price, from [tex]\( x = 0 \)[/tex] to [tex]\( x = 250 \)[/tex]. This can be calculated as:
[tex]\[
\text{Consumer Surplus} = \int_{0}^{250} [d(x) - \text{Equilibrium Price}] \, dx
\][/tex]
3. Set up the integral:
[tex]\[
\int_{0}^{250} [200 - 0.6x - 50] \, dx = \int_{0}^{250} [150 - 0.6x] \, dx
\][/tex]
4. Evaluate the integral:
First, compute the integral:
[tex]\[
\int_{0}^{250} [150 - 0.6x] \, dx = \left[ 150x - 0.3x^2 \right]_{0}^{250}
\][/tex]
Plug in the limits:
[tex]\[
\left( 150 \times 250 - 0.3 \times 250^2 \right) - \left( 150 \times 0 - 0.3 \times 0^2 \right)
\][/tex]
Calculate each term:
[tex]\[
150 \times 250 = 37500
\][/tex]
[tex]\[
0.3 \times 250^2 = 0.3 \times 62500 = 18750
\][/tex]
So, the consumer surplus is:
[tex]\[
37500 - 18750 = 18750
\][/tex]
Therefore, the equilibrium quantity is [tex]\( \boxed{250} \)[/tex] and the consumer surplus at the equilibrium quantity is [tex]\( \boxed{18750} \)[/tex].
