Question

Katya is a ranger at a nature reserve in Siberia. Russia. where she studies the changes in the reserve's beaı

population over time.
The relationship between the elapsed time t, in years, since the beginning of the study and the bear population
B(t) , on the reserve is modeled by the following function.
B(t)=5000· 2⁻⁰.⁰⁵ᵗ
In how many years will the reserve's bear population be 2000?
Round your answer, if necessary, to the nearest hundredth.
_____years

Answer 1

Katya's bear population model predicts that it will take approximately 13.29 years for the bear population to decrease to 2000 at the given decay rate.

Step-by-step explanation:

The problem presented involves an exponential decay model for a bear population in a nature reserve in Siberia, Russia. Katya, a ranger at the reserve, must determine the time it will take for the bear population to decrease to 2000 from an initial population of 5000. The function B(t)=5000\u00b7 2^{-0.05t} models the decay of the bear population over time, where t is time in years, and B(t) is the bear population at time t.

To solve for the time when the population reaches 2000, we set B(t)=2000 and solve for t:

1. Start with the equation 2000 = 5000 \u00b7 2^{-0.05t}.
2. Divide both sides by 5000 to isolate the exponential term: 0.4 = 2^{-0.05t}.
3. Apply the logarithm to both sides, using the base of 2, since our equation involves a base 2 exponential: log2(0.4) = -0.05t.
4. Solve for t by dividing both sides by -0.05: t = log2(0.4) / -0.05.
5. Calculate the value to determine the number of years: t ≈ 13.29 years.

Therefore, it will take approximately 13.29 years for the bear population to decrease to 2000 bears.