Let's find the profit function [tex]\( p(x) \)[/tex] step-by-step, starting from the given revenue and cost functions.
1. Revenue function:
[tex]\[
r(x) = 15x
\][/tex]
This function denotes the revenue obtained from selling [tex]\( x \)[/tex] shirts, where each shirt sells for [tex]$15.
2. Cost function:
\[
c(x) = 7x + 20
\]
This function represents the total cost of buying \( x \) shirts, with each shirt costing $[/tex]7 and an additional fixed cost of $20.
3. Profit function:
The profit function [tex]\( p(x) \)[/tex] is the revenue function [tex]\( r(x) \)[/tex] minus the cost function [tex]\( c(x) \)[/tex]. Mathematically, this can be expressed as:
[tex]\[
p(x) = r(x) - c(x)
\][/tex]
Substitute the expressions for [tex]\( r(x) \)[/tex] and [tex]\( c(x) \)[/tex]:
[tex]\[
p(x) = 15x - (7x + 20)
\][/tex]
4. Simplifying the expression:
Distribute the negative sign across the terms inside the parentheses:
[tex]\[
p(x) = 15x - 7x - 20
\][/tex]
Combine like terms:
[tex]\[
p(x) = (15x - 7x) - 20
\][/tex]
[tex]\[
p(x) = 8x - 20
\][/tex]
Thus, the profit function [tex]\( p(x) \)[/tex] is:
[tex]\[
p(x) = 8x - 20
\][/tex]
The correct answer is:
C. [tex]\( p(x) = 8x - 20 \)[/tex]
