The number of mosquitoes in Minneapolis, Minnesota (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by:

Question

$$m(x)=-(x-5{)}^2+25$$
How many centimeters of rainfall will produce the maximum number of mosquitoes?
centimeters

Answer 1

To determine the amount of rainfall that produces the maximum number of mosquitoes according to the given function $m(x)=-(x-5{)}^2+25$, we first recognize that this is a quadratic function in the standard form $m(x)=a(x-h{)}^2+k$, where $(h,k)$ is the vertex of the parabola.

In this case:

• The coefficient $a=-1$ indicates that the parabola opens downwards, which means there is a maximum point.
• The expression $(x-5{)}^2$ suggests that the vertex (which provides $h$) is at $x=5$.

Now, we know that $h=5$ is the $x$-value at which the maximum occurs. The maximum number of mosquitoes will occur when the rainfall is:

$$\boxed{{\displaystyle 5}}$$

centimeters.