To determine the profit function \( p(x) \) from selling \( x \) shirts, we must first understand and work with the given expressions for revenue and cost:
1. Revenue Function:
The revenue from selling \( x \) shirts is given by:
[tex]\[
r(x) = 15x
\][/tex]
2. Cost Function:
The cost of buying \( x \) shirts is given by:
[tex]\[
c(x) = 7x + 20
\][/tex]
3. Profit Function:
The profit is the revenue minus the cost. Therefore, the profit function \( p(x) \) is defined as:
[tex]\[
p(x) = r(x) - c(x)
\][/tex]
Let's substitute the expressions for \( r(x) \) and \( c(x) \) into the profit function:
[tex]\[
p(x) = 15x - (7x + 20)
\][/tex]
Next, we need to simplify the expression inside the parentheses:
[tex]\[
p(x) = 15x - 7x - 20
\][/tex]
Subtract \( 7x \) from \( 15x \):
[tex]\[
p(x) = 8x - 20
\][/tex]
So, the profit function \( p(x) \) is:
[tex]\[
p(x) = 8x - 20
\][/tex]
Therefore, the correct answer is:
D. [tex]\( p(x) = 8x - 20 \)[/tex]
