Answer :
To find the profit function [tex]\( p(x) \)[/tex], we start by understanding the given functions:
1. Revenue Function: The revenue [tex]\( r(x) \)[/tex] from selling [tex]\( x \)[/tex] shirts is given by the equation:
[tex]\[ r(x) = 15x \][/tex]
This means that each shirt is sold for [tex]$15. 2. Cost Function: The cost \( c(x) \) of buying \( x \) shirts is given by the equation: \[ c(x) = 7x + 20 \] This indicates that each shirt costs $[/tex]7 to buy, plus an additional fixed cost of $20, possibly representing overhead or initial cost.
3. Profit Function: The profit [tex]\( p(x) \)[/tex] is defined as the difference between the revenue and the cost. Therefore:
[tex]\[ p(x) = r(x) - c(x) \][/tex]
Now, substitute the given functions into the profit equation:
[tex]\[ p(x) = (15x) - (7x + 20) \][/tex]
Next, simplify the equation by distributing and combining like terms:
[tex]\[ p(x) = 15x - 7x - 20 \][/tex]
[tex]\[ p(x) = 8x - 20 \][/tex]
Therefore, the profit function [tex]\( p(x) \)[/tex] is:
[tex]\[ \boxed{8x - 20} \][/tex]
1. Revenue Function: The revenue [tex]\( r(x) \)[/tex] from selling [tex]\( x \)[/tex] shirts is given by the equation:
[tex]\[ r(x) = 15x \][/tex]
This means that each shirt is sold for [tex]$15. 2. Cost Function: The cost \( c(x) \) of buying \( x \) shirts is given by the equation: \[ c(x) = 7x + 20 \] This indicates that each shirt costs $[/tex]7 to buy, plus an additional fixed cost of $20, possibly representing overhead or initial cost.
3. Profit Function: The profit [tex]\( p(x) \)[/tex] is defined as the difference between the revenue and the cost. Therefore:
[tex]\[ p(x) = r(x) - c(x) \][/tex]
Now, substitute the given functions into the profit equation:
[tex]\[ p(x) = (15x) - (7x + 20) \][/tex]
Next, simplify the equation by distributing and combining like terms:
[tex]\[ p(x) = 15x - 7x - 20 \][/tex]
[tex]\[ p(x) = 8x - 20 \][/tex]
Therefore, the profit function [tex]\( p(x) \)[/tex] is:
[tex]\[ \boxed{8x - 20} \][/tex]
