Katya is a ranger at a nature reserve in Siberia, Russia, where she studies the changes in the reserve's bear population over time.
The relationship between the elapsed time [tex]$t$[/tex], in years, since the beginning of the study and the bear population [tex]$B(t)$[/tex], on the reserve is modeled by the following function.
[tex]$B(t)=5000 \cdot 2^{-0.05 t}$[/tex]
In how many years will the reserve's bear population be 2000? Round your answer, if necessary, to the nearest hundredth.
[ ] years
Answer (1)
Set up the equation: 2000 = 5000 c d o t 2 − 0.05 t .
Simplify the equation: 0.4 = 2 − 0.05 t .
Take the logarithm of both sides and solve for t : t = − 0.05 l n ( 2 ) l n ( 0.4 ) .
Calculate t and round to the nearest hundredth: t ≈ 26.44 years.
Explanation
Understanding the Problem We are given the function B ( t ) = 5000 c d o t 2 − 0.05 t that models the bear population B ( t ) after t years. We want to find the time t when the bear population is 2000.
Setting up the Equation We set B ( t ) = 2000 and solve for t . This gives us the equation 2000 = 5000 c d o t 2 − 0.05 t .
Simplifying the Equation Divide both sides of the equation by 5000: 5000 2000 = 2 − 0.05 t which simplifies to 0.4 = 2 − 0.05 t .
Taking the Logarithm Take the logarithm of both sides. We can use any base for the logarithm, but we'll use the natural logarithm (base e ): ln ( 0.4 ) = ln ( 2 − 0.05 t ) .
Applying Logarithm Properties Use the logarithm property ln ( a b ) = b ln ( a ) : ln ( 0.4 ) = − 0.05 t ln ( 2 ) .
Isolating t Solve for t : t = − 0.05 ln ( 2 ) ln ( 0.4 ) .
Calculating t Using a calculator, we find that t ≈ − 0.05 × 0.69315 − 0.91629 ≈ − 0.0346575 − 0.91629 ≈ 26.43856 . Rounding to the nearest hundredth, we get t ≈ 26.44 years.
Final Answer Therefore, it will take approximately 26.44 years for the bear population to be 2000.
Examples
Exponential decay models, like the one describing the bear population, are useful in various real-world scenarios. For instance, they can model the depreciation of a car's value over time, the decay of radioactive substances in nuclear medicine, or the cooling of a hot beverage. Understanding exponential decay helps in making informed decisions about investments, medical treatments, and even everyday tasks like brewing coffee.
