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The fox population in a certain region has an annual growth rate of 7 percent per year. It is estimated that the population in the year 2000 was 24200. (a) Find a function that models the population t years after 2000 (t = 0 for 2000). %3D Your answer is P(t) (b) Use the function from part (a) to estimate the fox population in the year 2008. Your answer is (the answer should be an integer)

The fox population in a certain region has an annual growth rate of 7 percent per year. It is estimated that the population in the year 2000 was 24200. (a) Find a function that models the population t years after 2000 (t = 0 for 2000). %3D Your answer is P(t) (b) Use the function from part (a) to estimate the fox population in the year 2008. Your answer is (the answer should be an integer)

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Population Growth Problem Set #### Problem Statement: The fox population in a certain region has an annual growth rate of 7 percent per year. It is estimated that the population in the year 2000 was 24,200. #### Tasks: 1. **Find a Function That Models the Population** - Goal: Develop a mathematical function that represents the fox population \( t \) years after the year 2000 (\( t = 0 \) for the year 2000). - Answer format: \( P(t) = \) __________ 2. **Estimate the Fox Population in 2008 Using the Function from Part (a)** - Goal: Use the developed function to estimate the fox population in the year 2008. - Answer format: The population estimate should be an integer. - Answer format: The answer is (the answer should be an integer) __________ #### Solution: (a) **Find a Function that Models the Population** To model the fox population, we use the formula for exponential growth: \[ P(t) = P_0 (1 + r)^t \] Where: - \( P(t) \) is the population at time \( t \), - \( P_0 \) is the initial population (24,200 in the year 2000), - \( r \) is the annual growth rate (7% or 0.07), - \( t \) is the number of years after 2000. So the function that models the population \( t \) years after 2000 is: \[ P(t) = 24200 \times (1 + 0.07)^t \] (b) **Estimate the Fox Population in the Year 2008** For the year 2008, \( t = 2008 - 2000 = 8 \). Using the function from part (a): \[ P(8) = 24200 \times (1.07)^8 \] Calculate \( (1.07)^8 \): \[ (1.07)^8 \approx 1.718186 \] Now multiply by the initial population: \[ P(8) \approx 24200 \times 1.718186 \approx 41580 \] So, the estimated fox population in the year 2008 is approximately 41,580. **Your answer is:** \[
Transcribed Image Text:### Population Growth Problem Set #### Problem Statement: The fox population in a certain region has an annual growth rate of 7 percent per year. It is estimated that the population in the year 2000 was 24,200. #### Tasks: 1. **Find a Function That Models the Population** - Goal: Develop a mathematical function that represents the fox population \( t \) years after the year 2000 (\( t = 0 \) for the year 2000). - Answer format: \( P(t) = \) __________ 2. **Estimate the Fox Population in 2008 Using the Function from Part (a)** - Goal: Use the developed function to estimate the fox population in the year 2008. - Answer format: The population estimate should be an integer. - Answer format: The answer is (the answer should be an integer) __________ #### Solution: (a) **Find a Function that Models the Population** To model the fox population, we use the formula for exponential growth: \[ P(t) = P_0 (1 + r)^t \] Where: - \( P(t) \) is the population at time \( t \), - \( P_0 \) is the initial population (24,200 in the year 2000), - \( r \) is the annual growth rate (7% or 0.07), - \( t \) is the number of years after 2000. So the function that models the population \( t \) years after 2000 is: \[ P(t) = 24200 \times (1 + 0.07)^t \] (b) **Estimate the Fox Population in the Year 2008** For the year 2008, \( t = 2008 - 2000 = 8 \). Using the function from part (a): \[ P(8) = 24200 \times (1.07)^8 \] Calculate \( (1.07)^8 \): \[ (1.07)^8 \approx 1.718186 \] Now multiply by the initial population: \[ P(8) \approx 24200 \times 1.718186 \approx 41580 \] So, the estimated fox population in the year 2008 is approximately 41,580. **Your answer is:** \[
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