Scientists studied a deer population for 10 years. They generated the function [tex]$f(x)=248(1.15)^x$[/tex] to approximate the number of deer in the population [tex]$x$[/tex] years after beginning the study. About how many deer are in the population 3 years after beginning the study?
A. 251
B. 377
C. 856
D. 1,003
Answer (2)
Substitute x = 3 into the function f ( x ) = 248 ( 1.15 ) x .
Calculate f ( 3 ) = 248 ( 1.15 ) 3 .
Evaluate the expression: f ( 3 ) = 248 × 1.520875 = 377.177 .
Round the result to the nearest whole number: 377 .
Explanation
Understanding the Problem We are given the function f ( x ) = 248 ( 1.15 ) x which models the deer population x years after the study began. We want to find the population 3 years after the study began, so we need to find f ( 3 ) .
Substituting x=3 To find the deer population after 3 years, we substitute x = 3 into the function: f ( 3 ) = 248 ( 1.15 ) 3
Calculating the Value Now, we calculate ( 1.15 ) 3 which is approximately 1.520875 . Then we multiply this by 248: f ( 3 ) = 248 × 1.520875 = 377.177
Rounding the Result Since we are looking for an approximate number of deer, we round the result to the nearest whole number: f ( 3 ) ≈ 377 Therefore, the approximate number of deer in the population 3 years after the beginning of the study is 377.
Final Answer The approximate number of deer in the population 3 years after the beginning of the study is 377.
Examples
Population growth models, like the one used for the deer, are commonly used in biology and ecology. For example, scientists can use these models to predict the growth of bacteria in a culture, the spread of an invasive species in a new environment, or the recovery of an endangered species after conservation efforts. By understanding these growth patterns, researchers can make informed decisions about resource management, conservation strategies, and public health interventions. The exponential growth function helps in understanding how populations change over time and allows for predictions about future population sizes.
After substituting x=3 into the function and calculating, the approximate number of deer in the population 3 years after the study began is 377. Thus, the correct multiple choice option is B. 377.
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