the number of mosquitoes m(x), in millions, in a certain area depends on the june rainfall, x, in inches…

Answer:

$6.5$ inches

Explanation:

Step1: Identify the function type

$M(x)=13x - x^{2}$ is a quadratic function in the form $y = ax^{2}+bx + c$ where $a=- 1$, $b = 13$, $c = 0$.

Step2: Recall the formula for the vertex of a quadratic

The $x$-coordinate of the vertex of a quadratic function $y=ax^{2}+bx + c$ is given by $x=-\frac{b}{2a}$.

Step3: Substitute the values of a and b

Substitute $a=-1$ and $b = 13$ into the formula $x=-\frac{b}{2a}$. We get $x=-\frac{13}{2\times(-1)}=\frac{13}{2}=6.5$. This $x$-value gives the rainfall that produces the maximum number of mosquitoes since for a quadratic function with $a<0$, the vertex represents the maximum - point of the function.