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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Question
SEE THE FORMULA AND SOLVE
![**Sample:**
**Growth Function Formula**
\[ P(t) = P_0 e^{kt} \]
- \( P(t) \) represents the population at time \( t \).
- \( P_0 \) indicates the initial value of the population.
- \( k \) is the constant growth rate.
- \( t \) denotes time.
This formula is commonly used to model exponential growth, such as population or investment growth over time. The exponential term \( e^{kt} \) reflects how the quantity grows at a rate proportional to its current value, illustrating the continuous and constant percentage growth.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b8d1efa-c07b-4629-8963-670ddc0d1c0b%2F0619d003-0418-4bf7-9369-350ad4479514%2Fdjghgp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Sample:**
**Growth Function Formula**
\[ P(t) = P_0 e^{kt} \]
- \( P(t) \) represents the population at time \( t \).
- \( P_0 \) indicates the initial value of the population.
- \( k \) is the constant growth rate.
- \( t \) denotes time.
This formula is commonly used to model exponential growth, such as population or investment growth over time. The exponential term \( e^{kt} \) reflects how the quantity grows at a rate proportional to its current value, illustrating the continuous and constant percentage growth.
![### Exponential Bacterial Growth Problem
**Problem Statement:**
A bacteria culture that exhibits exponential growth doubles in size in 10 days.
**Objective:**
Find the growth constant.
---
### Solution Guide:
1. **Understanding the Problem:**
- Exponential growth can be represented by the equation:
\[
N(t) = N_0 e^{kt}
\]
where:
- \(N(t)\) is the population at time \(t\),
- \(N_0\) is the initial population,
- \(k\) is the growth constant,
- \(t\) is time.
2. **Given Information:**
- The bacteria culture doubles in size in 10 days.
- If \(N_0\) is the initial population, then after 10 days (\(t=10\)):
\[
N(10) = 2N_0
\]
3. **Setting Up the Equation:**
\[
2N_0 = N_0 e^{k \cdot 10}
\]
4. **Solving for the Growth Constant (\(k\)):**
- Divide both sides of the equation by \(N_0\):
\[
2 = e^{10k}
\]
- Take the natural logarithm of both sides:
\[
\ln(2) = 10k
\]
- Isolate \(k\):
\[
k = \frac{\ln(2)}{10}
\]
5. **Answer:**
- The growth constant \(k\) is:
\[
k = \frac{\ln(2)}{10} \approx 0.0693
\]
By solving the provided problem in this manner, you can determine the exponential growth constant for the bacteria culture based on the given information.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b8d1efa-c07b-4629-8963-670ddc0d1c0b%2F0619d003-0418-4bf7-9369-350ad4479514%2F63v1ds_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Exponential Bacterial Growth Problem
**Problem Statement:**
A bacteria culture that exhibits exponential growth doubles in size in 10 days.
**Objective:**
Find the growth constant.
---
### Solution Guide:
1. **Understanding the Problem:**
- Exponential growth can be represented by the equation:
\[
N(t) = N_0 e^{kt}
\]
where:
- \(N(t)\) is the population at time \(t\),
- \(N_0\) is the initial population,
- \(k\) is the growth constant,
- \(t\) is time.
2. **Given Information:**
- The bacteria culture doubles in size in 10 days.
- If \(N_0\) is the initial population, then after 10 days (\(t=10\)):
\[
N(10) = 2N_0
\]
3. **Setting Up the Equation:**
\[
2N_0 = N_0 e^{k \cdot 10}
\]
4. **Solving for the Growth Constant (\(k\)):**
- Divide both sides of the equation by \(N_0\):
\[
2 = e^{10k}
\]
- Take the natural logarithm of both sides:
\[
\ln(2) = 10k
\]
- Isolate \(k\):
\[
k = \frac{\ln(2)}{10}
\]
5. **Answer:**
- The growth constant \(k\) is:
\[
k = \frac{\ln(2)}{10} \approx 0.0693
\]
By solving the provided problem in this manner, you can determine the exponential growth constant for the bacteria culture based on the given information.
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