Let's walk through this problem step by step to find Luisa's profit function [tex]\( p(x) \)[/tex].
1. Determine the revenue function:
Luisa's revenue [tex]\( r(x) \)[/tex] for mowing [tex]\( x \)[/tex] lawns is given by:
[tex]\[
r(x) = 20x
\][/tex]
2. Determine the cost function:
The cost [tex]\( c(x) \)[/tex] for gas and mower rental for mowing [tex]\( x \)[/tex] lawns is given by:
[tex]\[
c(x) = 4x + 25
\][/tex]
3. Derive the profit function:
The profit [tex]\( p(x) \)[/tex] is the difference between the revenue and the costs, which can be written as:
[tex]\[
p(x) = r(x) - c(x)
\][/tex]
4. Substitute the revenue and cost functions into the profit function:
[tex]\[
p(x) = 20x - (4x + 25)
\][/tex]
5. Simplify the equation:
[tex]\[
p(x) = 20x - 4x - 25
\][/tex]
[tex]\[
p(x) = 16x - 25
\][/tex]
Therefore, the profit function [tex]\( p(x) \)[/tex] is:
[tex]\[
p(x) = 16x - 25
\][/tex]
Comparing this result with the given choices, we see that:
- A. [tex]\( p(x) = 16x - 25 \)[/tex]
- B. [tex]\( p(x) = 16x + 25 \)[/tex]
- C. [tex]\( p(x) = 24x - 25 \)[/tex]
- D. [tex]\( p(x) = 24x + 25 \)[/tex]
The correct answer is:
[tex]\[
\boxed{1}
\][/tex]
