The number of mosquitoes M(x), in millions, in a certain area depends on the June rainfall x, in inches: $M ( x ) = 16 x - x ^ { 2 }$ . What rainfall produces the maximum number of mosquitoes?
A) 16 in.
B) 256 in.
C) 0 in.
D) 8 in.

Question

Quizplus · Accepted Answer

The maximum value of a quadratic function $f(x) = ax^2 + bx + c$ occurs at $x = -\frac{b}{2a}$. For $M(x) = 16x - x^2$, we can rewrite it as $-x^2 + 16x$, making $a = -1$ and $b = 16$. Thus, the maximum occurs at $x = -\frac{16}{2(-1)} = 8$.

The Number of Mosquitoes M(x), in Millions, in a Certain M(x)=16x−x2M ( x ) = 16 x - x ^ { 2 }M(x)=16x−x2

The Number of Mosquitoes M(x), in Millions, in a Certain $M ( x ) = 16 x - x ^ { 2 }$