Answer :
To determine Mario's profit [tex]\( p(x) \)[/tex] from mowing [tex]\( x \)[/tex] lawns, we need to use the given revenue function [tex]\( r(x) \)[/tex] and cost function [tex]\( c(x) \)[/tex]:
1. The revenue function is [tex]\( r(x) = 25x \)[/tex].
2. The cost function is [tex]\( c(x) = 6x + 15 \)[/tex].
Profit [tex]\( p(x) \)[/tex] is the difference between revenue [tex]\( r(x) \)[/tex] and cost [tex]\( c(x) \)[/tex]:
[tex]\[ p(x) = r(x) - c(x) \][/tex]
We will substitute the given functions for revenue and cost into this formula:
[tex]\[ p(x) = 25x - (6x + 15) \][/tex]
Next, distribute the negative sign across the terms inside the parentheses:
[tex]\[ p(x) = 25x - 6x - 15 \][/tex]
Combine the like terms [tex]\( 25x \)[/tex] and [tex]\( -6x \)[/tex]:
[tex]\[ p(x) = (25x - 6x) - 15 \][/tex]
[tex]\[ p(x) = 19x - 15 \][/tex]
Thus, Mario's profit from mowing [tex]\( x \)[/tex] lawns is:
[tex]\[ p(x) = 19x - 15 \][/tex]
The correct answer is:
[tex]\[ \boxed{C. \, p(x) = 19x - 15} \][/tex]
1. The revenue function is [tex]\( r(x) = 25x \)[/tex].
2. The cost function is [tex]\( c(x) = 6x + 15 \)[/tex].
Profit [tex]\( p(x) \)[/tex] is the difference between revenue [tex]\( r(x) \)[/tex] and cost [tex]\( c(x) \)[/tex]:
[tex]\[ p(x) = r(x) - c(x) \][/tex]
We will substitute the given functions for revenue and cost into this formula:
[tex]\[ p(x) = 25x - (6x + 15) \][/tex]
Next, distribute the negative sign across the terms inside the parentheses:
[tex]\[ p(x) = 25x - 6x - 15 \][/tex]
Combine the like terms [tex]\( 25x \)[/tex] and [tex]\( -6x \)[/tex]:
[tex]\[ p(x) = (25x - 6x) - 15 \][/tex]
[tex]\[ p(x) = 19x - 15 \][/tex]
Thus, Mario's profit from mowing [tex]\( x \)[/tex] lawns is:
[tex]\[ p(x) = 19x - 15 \][/tex]
The correct answer is:
[tex]\[ \boxed{C. \, p(x) = 19x - 15} \][/tex]
