Question

the mosquito population is a function of rainfall, and can be approximated by the formula n(x)=1000 + 45x^2 - x^3 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. a) 15 b) 30 c) 5 d) 89/3 e) 31 f) none of the above.

Explanation:

Step1: Find the derivative of \(N(x)\)

We have \(N(x)=1000 + 45x^{2}-x^{3}\). Using the power - rule \((x^n)^\prime=nx^{n - 1}\), the derivative \(N^\prime(x)=90x-3x^{2}\).

Step2: Set the derivative equal to zero

To find the critical points, we set \(N^\prime(x) = 0\). So, \(90x-3x^{2}=0\). Factor out \(3x\): \(3x(30 - x)=0\).

Step3: Solve for \(x\)

From \(3x(30 - x)=0\), we get two solutions: \(3x=0\) gives \(x = 0\), and \(30 - x=0\) gives \(x = 30\).

Step4: Use the second - derivative test

Find the second - derivative \(N^{\prime\prime}(x)=90 - 6x\).
When \(x = 0\), \(N^{\prime\prime}(0)=90>0\), so the function has a local minimum at \(x = 0\).
When \(x = 30\), \(N^{\prime\prime}(30)=90-6\times30=90 - 180=- 90<0\), so the function has a local maximum at \(x = 30\).

Answer:

b. 30