Question

The number of mosquitoes M(x), in millions, in a certain area depends on the June rainfall x, in inches: M(x) = 13 x - x2. What rainfall produces the maximum number of mosquitoes

Answer 1

Answer:

The number of mosquitoes is maximum for 6.5 inches rainfall.

Step-by-step explanation:

The given function is

$M(x)=13x-x^2$                   .... (1)

where, M(x) is number of mosquitoes in millions and x the June rainfall in inches.

We need to find the rainfall that produces the maximum number of mosquitoes.

Differential the above function with respect to x.

$M'(x)=13-2x$               .... (2)

Equate first derivative equal to 0.

$13-2x=0$

$2x=13$

$x=\frac{13}{2}=6.5$

Differential function (2) with respect to x.

$M''(x)=-2$

Double derivative is negative. So, the value of function is maximum at x=6.5.

Therefore, the number of mosquitoes is maximum for 6.5 inches rainfall.