The number of mosquitoes M(x), in millions, in a certain area depends on the June rainfall x, in inches, according to the equation $M ( x ) = 20 x - x ^ { 2 }$ . What rainfall produces the maximum number of mosquitoes?
A) 20 in.
B) 400 in.
C) 10 in.
D) 0 in.

Question

Quizplus · Accepted Answer

The equation $M(x) = 20x - x^2$ is a quadratic equation in the form of $ax^2 + bx + c$, where $a = -1$, $b = 20$, and $c = 0$. The vertex of a parabola given by a quadratic equation $y = ax^2 + bx + c$ is found using the formula $x = -\frac{b}{2a}$. Substituting $a = -1$ and $b = 20$ into this formula gives $x = -\frac{20}{2(-1)} = 10$. Therefore, the rainfall that produces the maximum number of mosquitoes is 10 inches.

The Number of Mosquitoes M(x), in Millions, in a Certain M(x)=20x−x2M ( x ) = 20 x - x ^ { 2 }M(x)=20x−x2

The Number of Mosquitoes M(x), in Millions, in a Certain $M ( x ) = 20 x - x ^ { 2 }$