Question

The rabbit population on a small island is observed to be given by the function P(t) = 130t − 0.4t^4 + 1200 where t is the time (in months) since observations of the island began.

(a) When is the maximum population attained (Round your answer to one decimal place.)

What is the maximum population? (Round your answer to the nearest whole number.)

(b) When does the rabbit population disappear from the island? (Round your answer to one decimal place.) ...?

Answer 1

Explanation:

P(t)=130t -0.4t^4 +1200

The population will be max when first differential of p(t) =0

So p'(t) =130-1.6t^3=0

1.6t^3=130

t^3 =130/1.6

t^3 =81.25

t = cube root of 81.25

t =4.3 months

P(max) =130(4.3) -0.4(4.3)^4 +1200

= 559-136.75+1200

=1622

Population will disappear when p(t) =0

Answer 2

a.) P(t) = 130t - 0.4t^4 + 1200
The population is maximum when P'(t) = 0
P'(t) = 130 - 1.6t^3 = 0
1.6t^3 = 130
t^3 = 81.25
t = ∛81.25 = 4.3 months.

Maximum population P(t)max = 130(4.3) - 0.4(4.3)^4 + 1200 = 1,622

b.) The rabbit population will disappear when P(t) = 0
P(t) = 130t - 0.4t^4 + 1200 = 0
t ≈ 8.7 months