To determine the profit function [tex]\( p(x) \)[/tex], we need to find the difference between the revenue function [tex]\( r(x) \)[/tex] and the cost function [tex]\( c(x) \)[/tex].
First, let's explicitly write down the given functions:
- The revenue function is [tex]\( r(x) = 18x \)[/tex].
- The cost function is [tex]\( c(x) = 8x + 30 \)[/tex].
The profit function [tex]\( p(x) \)[/tex] is defined as the revenue minus the cost. Therefore, we have:
[tex]\[
p(x) = r(x) - c(x)
\][/tex]
Substitute the given functions into the equation:
[tex]\[
p(x) = 18x - (8x + 30)
\][/tex]
Next, distribute the negative sign inside the parentheses:
[tex]\[
p(x) = 18x - 8x - 30
\][/tex]
Combine like terms:
[tex]\[
p(x) = (18x - 8x) - 30
\][/tex]
[tex]\[
p(x) = 10x - 30
\][/tex]
Therefore, the profit function [tex]\( p(x) \)[/tex] is:
[tex]\[
p(x) = 10x - 30
\][/tex]
Among the given options, the correct answer is:
C. [tex]\( p(x) = 10x - 30 \)[/tex]
