Luisa earns money mowing her neighbors' lawns.

The revenue for mowing [tex]$x$[/tex] lawns is [tex]$r(x)=20x$[/tex].

Luisa's cost for gas and the mower rental is [tex][tex]$c(x)=4x+25$[/tex][/tex].

Her profit from mowing [tex]$x$[/tex] lawns is [tex]$p(x)=(r-c)(x)$[/tex]. What is [tex][tex]$p(x)$[/tex][/tex]?

A. [tex]$p(x)=16x-25$[/tex]
B. [tex]$p(x)=16x+25$[/tex]
C. [tex][tex]$p(x)=24x-25$[/tex][/tex]
D. [tex]$p(x)=24x+25$[/tex]



Answer :

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]