The function [tex][tex]$p(x) = -2(x-9)^2 + 100$[/tex][/tex] is used to determine the profit on T-shirts sold for [tex][tex]$x$[/tex][/tex] dollars.

What would the profit from sales be if the price of the T-shirts were [tex][tex]$\$[/tex]15[tex]$[/tex] apiece?

A. [tex]$[/tex]\[tex]$15$[/tex][/tex]
B. [tex][tex]$\$[/tex]28[tex]$[/tex]
C. [tex]$[/tex]\[tex]$172$[/tex][/tex]
D. [tex][tex]$\$[/tex]244$[/tex]



Answer :

To determine the profit we would earn from selling T-shirts at a price of [tex]$15 each, we need to substitute \( x = 15 \) into the profit function \( p(x) = -2(x-9)^2 + 100 \). Here are the steps: 1. Start with the given profit function: \[ p(x) = -2(x-9)^2 + 100 \] 2. Substitute \( x = 15 \) into the function: \[ p(15) = -2(15-9)^2 + 100 \] 3. Calculate the expression inside the parentheses first: \[ 15 - 9 = 6 \] 4. Now square the result: \[ 6^2 = 36 \] 5. Multiply this squared result by \(-2\): \[ -2 \times 36 = -72 \] 6. Finally, add 100 to -72 to find the profit: \[ -72 + 100 = 28 \] So, the profit when the T-shirts are sold for $[/tex]15 each would be \$28. Thus, the correct answer is:

[tex]\(\boxed{28}\)[/tex]