a company models its revenue as $r(x)=30x - 2x^{2}$ and expenses as $e(x)=10x + 100$. determine the maximum…
a company models its revenue as $r(x)=30x - 2x^{2}$ and expenses as $e(x)=10x + 100$. determine the maximum profit.\n a. $x = 20$\n b. $x = 25$\n c. $x = 10$\n d. $x = 15$
Answer
- First, recall the profit - function formula:
- The profit function (P(x)) is given by (P(x)=R(x)-C(x)), where (R(x)) is the revenue function and (C(x)) is the cost (expense) function.
- Given (R(x) = 50x-2x^{2}) and (C(x)=10x + 100), then (P(x)=(50x-2x^{2})-(10x + 100)=-2x^{2}+40x - 100).
- For a quadratic function (y = ax^{2}+bx + c) ((a\neq0)), the (x) - coordinate of the vertex (which gives the maximum or minimum of the quadratic function) is (x=-\frac{b}{2a}).
- In the profit - function (P(x)=-2x^{2}+40x - 100), (a=-2), (b = 40), and (c=-100).
- Using the formula (x=-\frac{b}{2a}), we have (x=-\frac{40}{2\times(-2)}=\frac{-40}{-4}=10).
- Then, find the maximum profit by substituting (x = 10) into the profit function:
- (P(10)=-2\times(10)^{2}+40\times10-100=-2\times100 + 400-100=-200 + 400-100=100).
However, it seems the question is asking for the value of (x) (the quantity) that maximizes the profit.