To determine the profit from sales using the function [tex]\( p(x) = -2(x - 9)^2 + 100 \)[/tex], we need to substitute the price of the T-shirts into the function. The price of the T-shirts is given as [tex]\( x = 15 \)[/tex] dollars.
Here's the step-by-step process:
1. Identify the function:
[tex]\[
p(x) = -2(x - 9)^2 + 100
\][/tex]
2. Substitute the price into the function:
[tex]\[
p(15) = -2(15 - 9)^2 + 100
\][/tex]
3. Calculate the difference inside the parentheses:
[tex]\[
15 - 9 = 6
\][/tex]
4. Square the result:
[tex]\[
6^2 = 36
\][/tex]
5. Multiply by -2:
[tex]\[
-2 \times 36 = -72
\][/tex]
6. Add 100 to the result:
[tex]\[
-72 + 100 = 28
\][/tex]
Therefore, the profit from sales when the T-shirts are priced at \[tex]$15 apiece is \$[/tex]28.
