Absolutely, let's work through this step-by-step.
### Step 1: Determine the Demand Function
Given:
- The quantity of T-shirts sold at the equilibrium price ([tex]\( p = \$17 \)[/tex]) is 140.
- For each dollar decrease in price, 20 more T-shirts are sold.
We need to find a linear demand function of the form:
[tex]\[ q = ap + b \][/tex]
We know:
- When [tex]\( p = 17 \)[/tex], [tex]\( q = 140 \)[/tex]
- Each dollar decrease in price results in selling 20 additional T-shirts.
Using the second piece of information, we can determine the slope of the demand function:
[tex]\[ \text{slope} = -20 \text{ (since \( q \) increases by 20 as \( p \) decreases by 1)} \][/tex]
Therefore, the demand function is of the form:
[tex]\[ q = b - 20p \][/tex]
where [tex]\( b \)[/tex] is the intercept.
We substitute the known point ([tex]\( p = 17, q = 140 \)[/tex]) to find [tex]\( b \)[/tex]:
[tex]\[ 140 = b - 20 \cdot 17 \][/tex]
[tex]\[ 140 = b - 340 \][/tex]
[tex]\[ b = 480 \][/tex]
So, the demand function is:
[tex]\[ q = 480 - 20p \][/tex]
### Step 2: Find the Quantity Sold at [tex]\( p = \$13 \)[/tex]
To find the quantity sold when the price is [tex]\( p = 13 \)[/tex]:
[tex]\[ q = 480 - 20 \cdot 13 \][/tex]
[tex]\[ q = 480 - 260 \][/tex]
[tex]\[ q = 220 \][/tex]
### Step 3: Calculate the Consumer Surplus
Consumer surplus is the area between the demand curve and the price line, from the given price (\[tex]$13) up to the equilibrium price (\$[/tex]17).
First, the demand function:
[tex]\[ q = 480 - 20p \][/tex]
Consumer surplus can be calculated as:
[tex]\[ \text{Consumer Surplus} = \int_{p=13}^{17} (480 - 20p - 13) \, dp \][/tex]
Subtracting 13 from the demand function:
[tex]\[ \text{Function to integrate} = (480 - 20p - 13) = 467 - 20p \][/tex]
Now, we integrate [tex]\( 467 - 20p \)[/tex] with respect to [tex]\( p \)[/tex] from 13 to 17:
[tex]\[ \int_{13}^{17} (467 - 20p) \, dp \][/tex]
Integrate term-by-term:
[tex]\[ \int_{13}^{17} 467 \, dp - \int_{13}^{17} 20p \, dp \][/tex]
Compute the first integral:
[tex]\[ 467p \Big|_{13}^{17} = 467 \cdot 17 - 467 \cdot 13 = 7939 - 6071 = 1868 \][/tex]
Compute the second integral:
[tex]\[ 20 \int_{13}^{17} p \, dp = 20 \left[ \frac{p^2}{2} \right]_{13}^{17} = 20 \left[ \frac{17^2}{2} - \frac{13^2}{2} \right]
= 20 \left[ \frac{289}{2} - \frac{169}{2} \right]
= 20 \left[ \frac{120}{2} \right]
= 20 \cdot 60
= 1200 \][/tex]
Subtract the results of the two integrals:
[tex]\[ \text{Consumer Surplus} = 1868 - 1200 = 668 \][/tex]
Therefore, the consumer surplus when the shirts are sold for [tex]\( \$13 \)[/tex] each is:
[tex]\[ \boxed{668} \][/tex]
