To determine the monthly profit function [tex]\( p(w) \)[/tex] for Wacky Widgets, we need to subtract the cost function [tex]\( c(w) \)[/tex] from the revenue function [tex]\( r(w) \)[/tex].
Given:
- The revenue function [tex]\( r(w) = 90w \)[/tex]
- The cost function [tex]\( c(w) = 57w + 5,000 \)[/tex]
We know that profit [tex]\( p(w) \)[/tex] is defined as the revenue minus the cost:
[tex]\[ p(w) = r(w) - c(w) \][/tex]
Substitute the given functions into the profit equation:
[tex]\[ p(w) = 90w - (57w + 5,000) \][/tex]
Now, simplify the expression inside the parentheses:
[tex]\[ p(w) = 90w - 57w - 5,000 \][/tex]
Combine like terms:
[tex]\[ p(w) = (90 - 57)w - 5,000 \][/tex]
Perform the subtraction:
[tex]\[ p(w) = 33w - 5,000 \][/tex]
Therefore, the function that Wacky Widgets can use to calculate its monthly profit is:
[tex]\[ p(w) = 90w - (57w + 5,000) \][/tex]
The correct answer is:
[tex]\[ p(w) = 90w - (57w + 5,000) \][/tex]
