Population Growth Sea otter populations in Alaska have grown exponentially since their re-introduction in 1968. Initially, 400 sea otters were transplanted. In 1990, the population had reached 5,000.

Question

(a) What is the population of the sea otters in 2020?

(b) Find the current rate of change of this population.

Answer 1

If at 1968 time t = 0, then poulation growth model looking like this N = N₀e^rt, N₀ = 400, if t = 1990 - 1968 = 22;

N = 5000. Then we have equation for r: 5000 = 400e^22r; 5000/400 = e^22r; 22r = ln12.5; r = ln12.5/22 = .1148.

So N = 400e^.1148t. In 2020 t = 2020 - 1968 = 52

So in 2020 we have N = 400e^.1148·52= 156540

Answer 2

Part a)

expect exponential growth so eq 1: P=400e^rt

eq2: 5000=400e^r*12

use eq2 to find r = (1/12)*Ln(25/2)

use this r to plug into the general eq1 and use t=42

P=400e^7/2*(25/2)=5000e^7/2

Part b)

take the derivative and plug in t=42

dP/dt=5000e^t/12*(1/12)=(1250/3)e^t/12

plug in t=42 and dP/dt=(1250/3)e^7/2