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"." 2 0.01"."25 .
Calculate 2 0.25 ≈ 1.1892 .
Multiply by 2500: B ( 25 ) ≈ 2500"."1.1892 ≈ 2973.0177 .
Round to the nearest whole number: 2973 .
Explanation
Understanding the Problem We are given the function B ( t ) = 2500"." 2 0.01 t that models the black bear population B ( t ) in a national park after t years since the beginning of the study. We want to find the population in 25 years, so we need to calculate B ( 25 ) .
Substituting the Value of t To find the population in 25 years, we substitute t = 25 into the function: B ( 25 ) = 2500"." 2 0.01 ( 25 ) = 2500"." 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"." 2 0.25 ≈ 2500"."1.1892 ≈ 2973.0177
Rounding the Result Since we need to round the answer to the nearest whole number, we round 2973.0177 to 2973.
Final Answer Therefore, according to the model, the black bear population in the national park in 25 years will be approximately 2973 bears.
Examples
Understanding population growth models like the one used for the black bear population can be applied to various real-world scenarios. For instance, biologists use similar models to predict the growth of endangered species, helping them make informed conservation decisions. In business, these models can forecast market growth, allowing companies to plan their investments and strategies effectively. Even in public health, such models are used to track the spread of diseases and allocate resources to combat them.
The black bear population in the national park after 25 years is estimated to be approximately 2973 bears. This was calculated using the model B ( t ) = 2500 ⋅ 2 0.01 t by substituting t = 25 and evaluating the function. The final answer is rounded to the nearest whole number.
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