Enzo is studying the black bear population at a large national park.
He finds that the relationship between the elapsed time [tex]$t$[/tex], in years, since the beginning of the study, and the black bear population [tex]$B(t)$[/tex] , in the park is modeled by the following function.
[tex]$B(t)=2500 \cdot 2^{0.01 t}$[/tex]
According to the model, what will the black bear population be at that national park in 25 years?
Round your answer, if necessary, to the nearest whole number.
Answer (2)
Substitute t = 25 into the population model: B ( 25 ) = 2500 c d o t 2 0.01 c d o t 25 .
Simplify the exponent: B ( 25 ) = 2500 c d o t 2 0.25 .
Calculate the population: B ( 25 ) ≈ 2973.0177 .
Round to the nearest whole number: 2973 bears.
Explanation
Understanding the Problem We are given the function B ( t ) = 2500 c d o t 2 0.01 t that models the black bear population B ( t ) in a national park t years since the beginning of the study. We want to find the population in 25 years, which means we need to find B ( 25 ) .
Substituting the Value of t To find the black bear population in 25 years, we substitute t = 25 into the function: B ( 25 ) = 2500 c d o t 2 0.01 c d o t 25 B ( 25 ) = 2500 c d o t 2 0.25
Calculating the Population Now, we calculate 2 0.25 , which is the same as 2 4 1 or 4 2 . Then, we multiply the result by 2500. B ( 25 ) = 2500 c d o t 2 0.25 ≈ 2500 c d o t 1.1892 B ( 25 ) ≈ 2973.0177
Rounding the Result Since we need to round the answer to the nearest whole number, we round 2973.0177 to 2973. Therefore, the black bear population in 25 years will be approximately 2973 bears.
Examples
Understanding population growth is essential in wildlife management. For instance, park rangers can use population models like the one in this problem to predict the impact of environmental changes or conservation efforts on the black bear population. By estimating the population size after a certain period, they can make informed decisions about resource allocation, habitat preservation, and potential interventions to maintain a healthy and sustainable ecosystem. This proactive approach helps ensure the long-term well-being of the black bear population and the overall biodiversity of the national park.
After 25 years, the predicted black bear population in the national park is approximately 2973 bears. This is calculated using the model B ( t ) = 2500 ⋅ 2 0.01 t . The calculation involves substituting t = 25 into the model and simplifying the expression.
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