The number of mosquitoes, $M(x)$, in millions, in a certain area depends on the June rainfall, $x$, in inches. The function that models that phenomenon is $M(x)=12 x-x^{2}$. Find the amount of rainfall that will maximize the number of mosquitoes. What is the maximum number of mosquitoes?
What amount of rainfall will maximize the amount of mosquitoes?
inches
What is the maximum amount of mosquitoes?
million

Step 1 ：The number of mosquitoes, $M(x)$, in millions, in a certain area depends on the June rainfall, $x$, in inches. The function that models that phenomenon is $M(x)=12 x-x^{2}$. We need to find the amount of rainfall that will maximize the number of mosquitoes and the maximum number of mosquitoes.

Step 2 ：The maximum or minimum of a quadratic function occurs at its vertex. The x-coordinate of the vertex of a function in the form $f(x)=ax^{2}+bx+c$ is given by $-\frac{b}{2a}$. In this case, $a=-1$ and $b=12$, so the x-coordinate of the vertex is $-\frac{12}{2(-1)}=6$. This means that the amount of rainfall that will maximize the number of mosquitoes is 6 inches.

Step 3 ：To find the maximum number of mosquitoes, we substitute $x=6$ into the function $M(x)$.

Step 4 ：Final Answer: The amount of rainfall that will maximize the number of mosquitoes is \(\boxed{6}\) inches. The maximum number of mosquitoes is \(\boxed{36}\) million.