To determine which function represents the company's cost, [tex]\(C(x)\)[/tex], for producing lawnmowers, we will use the relationship between profit, revenue, and cost. The given functions are:
- Revenue: [tex]\(R(x) = 90x - x^2\)[/tex]
- Profit: [tex]\(P(x) = -x^2 + 30x - 200\)[/tex]
The relationship between these quantities is:
[tex]\[
\text{Profit} = \text{Revenue} - \text{Cost}
\][/tex]
Or symbolically:
[tex]\[
P(x) = R(x) - C(x)
\][/tex]
Rearranging this equation to solve for the cost function [tex]\(C(x)\)[/tex] gives:
[tex]\[
C(x) = R(x) - P(x)
\][/tex]
Now, substitute the given functions [tex]\(R(x)\)[/tex] and [tex]\(P(x)\)[/tex] into this equation:
[tex]\[
C(x) = (90x - x^2) - (-x^2 + 30x - 200)
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
C(x) = 90x - x^2 + x^2 - 30x + 200
\][/tex]
Combine the like terms:
[tex]\[
C(x) = (90x - 30x) + (-x^2 + x^2) + 200
\][/tex]
[tex]\[
C(x) = 60x + 200
\][/tex]
Thus, the cost function [tex]\(C(x)\)[/tex] is:
[tex]\[
C(x) = 60x + 200
\][/tex]
Therefore, the correct answer is:
C. [tex]\(C(x) = 60x + 200\)[/tex]
