Question

Katya is a ranger in a nature reserve in Siberia, Russia, where she studies the changes in the reserve's bear population size over time. She found that the number of bears, B, in the reserve at t years since the beginning of the study can be modeled by the following function: B(t)=5000*2^(-0.05t) How long will it take the bear population in the reserve to get to 2000 bears? Round your answer, if necessary, to 2 decimal places.

Answer 1

B(t) = 5000 * 2^(-0.05t)

2000 = 5000 * 2^(-0.05t) , div by 5000
0.4 = 2^(-0.05t) , ln on both sides =>

ln0.4 = (-0.05t)* ln2
(ln0.4)/ln2) /(-0.05) = t

t = 26.43
After 27 years

Answer 2

Answer: 26.44 years since the begginig of the study.

Explanation:

The populations of bears can be described by:

B(t) = 5000*2^(-0.05*t)

where t is in years.

we want to find t such:

5000*2^(-0.05*t) = 2000

2^(-0.05*t) = 2000/5000 = 2/5

then we use the rule:

a^y = x

then:

y = logₐ(x)

we use it in our equation and get:

2^(-0.05*t) = 2/5

-0.05*t = log₂(2/5) = ln(2/5)/ln(2)

t = ln(2/5)/ln(2) /-0.05 = 26.44

so 26.44 years since the begginig of the study.