The population in 2000 of a certain city was approximately 40,000. Assume the population is increasing at a rate of 8% per year. a. Write the exponential function that relates the total population as a function of t. Where t = 0 at the year 2000. P(t) : Preview b. Use a. to determine the rate at which the population is increasing in t years. Answer in the form of a · b’ where a is rounded to the nearest whole number. P'(t) = Preview
The population in 2000 of a certain city was approximately 40,000. Assume the population is increasing at a rate of 8% per year. a. Write the exponential function that relates the total population as a function of t. Where t = 0 at the year 2000. P(t) : Preview b. Use a. to determine the rate at which the population is increasing in t years. Answer in the form of a · b’ where a is rounded to the nearest whole number. P'(t) = Preview
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Exponential Growth and Population Increase
Given the following scenario:
- In the year 2000, a certain city's population was approximately 40,000.
- The population is assumed to be increasing at a rate of 8% per year.
#### Problem Statement
a) **Exponential Function for Population Growth**
Write the exponential function that relates the total population as a function of \( t \), where \( t = 0 \) at the year 2000.
\[ P(t) = \]
[Input Box] [Preview Button]
b) **Rate of Population Increase**
Use the function from part (a) to determine the rate at which the population is increasing in \( t \) years. Provide the answer in the form of \( a \cdot b^t \), where \( a \) is rounded to the nearest whole number.
\[ P'(t) = \]
[Input Box] [Preview Button]
c) **Population Increase After 5 Years**
Determine the rate at which the population is increasing in 5 years using the function from part (b). Round your answer to the nearest person.
\[ P'(5) = \]
[Input Box] \( \text{people per year} \) [Preview Button]
### Detailed Explanation of the Problem
- **Exponential Growth Formula**:
The standard form of an exponential growth function is given by:
\[
P(t) = P_0 \cdot e^{rt}
\]
Where:
- \( P(t) \) is the population at time \( t \).
- \( P_0 \) is the initial population (40,000 in the year 2000).
- \( r \) is the growth rate (8% or 0.08).
- \( t \) is the time in years since 2000.
- **Differentiating the Function**:
To find the rate of population change, differentiate \( P(t) \) with respect to \( t \).
\[
P'(t) = \frac{d}{dt} \left( P_0 \cdot e^{rt} \right)
\]
Applying the differentiation,
\[
P'(t) = P_0 \cdot r \cdot e^{rt}
\]
- **Calculating Specific Values**:
- For part (b), express](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8315635d-2d20-4360-9a0a-b8ebc2e330cb%2F0fbb9d7c-7419-4e38-906c-1d41ada1889a%2Fay7de8q_processed.png&w=3840&q=75)
Transcribed Image Text:### Exponential Growth and Population Increase
Given the following scenario:
- In the year 2000, a certain city's population was approximately 40,000.
- The population is assumed to be increasing at a rate of 8% per year.
#### Problem Statement
a) **Exponential Function for Population Growth**
Write the exponential function that relates the total population as a function of \( t \), where \( t = 0 \) at the year 2000.
\[ P(t) = \]
[Input Box] [Preview Button]
b) **Rate of Population Increase**
Use the function from part (a) to determine the rate at which the population is increasing in \( t \) years. Provide the answer in the form of \( a \cdot b^t \), where \( a \) is rounded to the nearest whole number.
\[ P'(t) = \]
[Input Box] [Preview Button]
c) **Population Increase After 5 Years**
Determine the rate at which the population is increasing in 5 years using the function from part (b). Round your answer to the nearest person.
\[ P'(5) = \]
[Input Box] \( \text{people per year} \) [Preview Button]
### Detailed Explanation of the Problem
- **Exponential Growth Formula**:
The standard form of an exponential growth function is given by:
\[
P(t) = P_0 \cdot e^{rt}
\]
Where:
- \( P(t) \) is the population at time \( t \).
- \( P_0 \) is the initial population (40,000 in the year 2000).
- \( r \) is the growth rate (8% or 0.08).
- \( t \) is the time in years since 2000.
- **Differentiating the Function**:
To find the rate of population change, differentiate \( P(t) \) with respect to \( t \).
\[
P'(t) = \frac{d}{dt} \left( P_0 \cdot e^{rt} \right)
\]
Applying the differentiation,
\[
P'(t) = P_0 \cdot r \cdot e^{rt}
\]
- **Calculating Specific Values**:
- For part (b), express
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