the mosquito population is a function of rainfall, and can be approximated by the formula n(x)=1000 + 45x^2…

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.

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.

Answer

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