The population of rabbits in the wilderness rises and falls as things like preda

Question

The population of rabbits in the wilderness rises and falls as things like predators and vegetation rise and fall. A study shows the population of rabbits (in hundred thousands) in a certain wilderness park is given by: P(t) = 4sin^2 (0.5t) + 1. where P is the population of the rabbits and t is the number of years that have passed since the study was started. Find the population of the rabbits when (i) t = 2 years (ii) t = 5 years. Find the rate of change of the number of rabbits with respect to time when (iii) t = 2 years (iv) t = 5 years (v) interpret the results of (iii) and (iv)

Explanation / Answer

Given equation

P(t)=4 sin^2 (0.5t) +1

(i) P(2)= 4 sin^2 (0.5 *2) +1= 4sin^2 (1) +1=3.832 x 10^5

(ii)P(5)=4 sin^2 (0.5 *5) +1= 4sin^2(2.5)+1=2.432 x 10^5

to find rate of change, we have to differentiate P(t)

P(t)=4 sin^2 (0.5t) +1

P '(t)= 4*2 sin(0.5t) cos(0.5t) (0.5)= 2sint

P '(2)= 2sin2=1.818 x 10^5. Population increasing at t=2.

P '(5)=2sin5=-1.917 x 10^5 Population decreasing at t=5

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