Answer :
To solve this problem, we need to determine the profit when T-shirts are sold at [tex]$15 each, using the given profit function \( p(x) = -2(x - 9)^2 + 100 \).
Here are the steps to find the profit:
1. Identify the given price \( x \):
- The price at which the T-shirts are sold is \( x = 15 \).
2. Substitute \( x = 15 \) into the profit function \( p(x) \) to calculate the profit:
\[
p(15) = -2(15 - 9)^2 + 100
\]
3. Simplify the expression inside the parentheses:
\[
15 - 9 = 6
\]
So the expression becomes:
\[
p(15) = -2(6)^2 + 100
\]
4. Calculate the square of 6:
\[
6^2 = 36
\]
5. Multiply by the coefficient (-2):
\[
-2 \times 36 = -72
\]
6. Add the constant term (100):
\[
p(15) = -72 + 100 = 28
\]
Therefore, the profit when T-shirts are sold at $[/tex]15 each is [tex]\( \$ 28 \)[/tex]. Hence, the correct answer is:
[tex]\[ \$ 28 \][/tex]
[tex]\[ \$ 28 \][/tex]
