Answer:
Therefore the population will be maximum when x=30 inches.
Step-by-step explanation:
Given that, the mosquito population is a function of rainfall and can be approximated by the formula
[tex]N(x)= -x^3+45x^2+1000[/tex]
where x is the number of inches of rainfall.
[tex]N(x)= -x^3+45x^2+1000[/tex]
Differentiating with respect to x
[tex]N'(x)= -3x^2+90x[/tex]
Again differentiating with respect to x
[tex]N''(x)=-6x+90[/tex]
Now we set N'(x)=0
[tex]-3x^2+90x=0[/tex]
[tex]\Rightarrow -3x(x-30)=0[/tex]
[tex]\Rightarrow x=0,30[/tex]
[tex]N''(0)=-6.0+90=90>0[/tex]
[tex]N''(30)=-6.30+90=-90<0[/tex]
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.
