Question

The mosquito population is a function of rainfall, and can be approximated by the formula N(x)=−x3+45x2+1000 where x is the number of inches of rainfall. Find the values of x (if any) so that the population will be a maximum? Note that x is non-negative.

Answer 1

Therefore the population will be maximum when x=30 inches.

Explanation:

Given that, the mosquito population is a function of rainfall and can be approximated by the formula

`N(x)= -x^3+45x^2+1000`

where x is the number of inches of rainfall.

`N(x)= -x^3+45x^2+1000`

Differentiating with respect to x

`N'(x)= -3x^2+90x`

Again differentiating with respect to x

`N''(x)=-6x+90`

Now we set N'(x)=0

`-3x^2+90x=0`

`\Rightarrow -3x(x-30)=0`

`\Rightarrow x=0,30`

`N''(0)=-6.0+90=90>0`

`N''(30)=-6.30+90=-90<0`

Since at x=30, N''(x)<0, So at x=30, N(x) has maximum value.

Therefore the population will be maximum when x=30 inches.