Sure, let's find Becca's profit function step-by-step.
1. Revenue Function \( r(x) \):
The revenue Becca earns from mowing \( x \) lawns is given by:
[tex]\[
r(x) = 18x
\][/tex]
This means Becca earns $18 per lawn she mows.
2. Cost Function \( c(x) \):
The cost for mowing \( x \) lawns, which includes gas and mower rental, is:
[tex]\[
c(x) = 5x + 20
\][/tex]
This means there is a fixed cost of [tex]$20 plus $[/tex]5 per lawn she mows.
3. Profit Function \( p(x) \):
Profit is the revenue minus the cost:
[tex]\[
p(x) = r(x) - c(x)
\][/tex]
Substituting in the given functions for \( r(x) \) and \( c(x) \):
[tex]\[
p(x) = 18x - (5x + 20)
\][/tex]
4. Simplify the Profit Function:
Distribute the negative sign through the cost function:
[tex]\[
p(x) = 18x - 5x - 20
\][/tex]
Combine like terms:
[tex]\[
p(x) = (18x - 5x) - 20
\][/tex]
[tex]\[
p(x) = 13x - 20
\][/tex]
Therefore, Becca's profit function \( p(x) \) is given by:
[tex]\[
p(x) = 13x - 20
\][/tex]
Using the multiple choices provided:
A. \( p(x) = 23x - 2 \)
B. \( p(x) = 23x + 20 \)
C. \( p(x) = 13x - 20 \)
D. \( p(x) = 13x + 20 \)
The correct answer is:
C. [tex]\( p(x) = 13x - 20 \)[/tex]
