#2. N(t)=No e# t Q2: If the yogurt is cultured for 10 hours according to this be model, what willthe ratio of the papulation to the initial population?
#2. N(t)=No e# t Q2: If the yogurt is cultured for 10 hours according to this be model, what willthe ratio of the papulation to the initial population?
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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![### Educational Content: Population Growth Model in Yogurt Culturing
#### Problem Statement
We are given a mathematical model to describe the population growth of bacteria in yogurt:
\[ N(t) = N_0 e^{\frac{t}{2} \cdot t} \]
**Question:** If the yogurt is cultured for 10 hours according to this model, what will be the ratio of the population to the initial population?
#### Explanation and Solution
##### Step 1: Understanding the Model
The given equation is an exponential growth model where:
- \( N(t) \) represents the population at time \( t \).
- \( N_0 \) is the initial population.
- \( t \) is the time in hours.
We need to find \( N(t)/N_0 \) after 10 hours.
##### Step 2: Finding the Growth Constant \( k \)
The standard form of the population model is:
\[ N(t) = N_0 e^{kt} \]
We are given that after 8 hours:
\[ \ln\left(\frac{N(8)}{N_0}\right) = 4 \]
This implies:
\[ \frac{N(8)}{N_0} = e^4 \]
Thus:
\[ N(8) = N_0 e^4 \]
This equation can also be expressed as \( N(8) = N_0 e^{8k} \).
Equating both expressions for \( N(8) \):
\[ N_0 e^4 = N_0 e^{8k} \]
Solving for \( k \):
\[ 4 = 8k \]
\[ k = \frac{4}{8} = \frac{1}{2} \]
Therefore, the growth constant \( k \) is \(\frac{1}{2}\).
##### Step 3: Calculating the Ratio after 10 Hours
Substitute \( k = \frac{1}{2} \) and \( t = 10 \) hours into the standard equation:
\[ N(t) = N_0 e^{\left(\frac{1}{2} \cdot 10\right)} \]
\[ N(10) = N_0 e^5 \]
The ratio of the population to the initial population after 10 hours is:
\[ \frac{N(10)}{N_0} = e^5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F113b0878-5af8-4998-be4b-de5d66497a2a%2F12abdf4f-5125-41f5-ba14-9fad39797b4e%2F5w2mbcd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Content: Population Growth Model in Yogurt Culturing
#### Problem Statement
We are given a mathematical model to describe the population growth of bacteria in yogurt:
\[ N(t) = N_0 e^{\frac{t}{2} \cdot t} \]
**Question:** If the yogurt is cultured for 10 hours according to this model, what will be the ratio of the population to the initial population?
#### Explanation and Solution
##### Step 1: Understanding the Model
The given equation is an exponential growth model where:
- \( N(t) \) represents the population at time \( t \).
- \( N_0 \) is the initial population.
- \( t \) is the time in hours.
We need to find \( N(t)/N_0 \) after 10 hours.
##### Step 2: Finding the Growth Constant \( k \)
The standard form of the population model is:
\[ N(t) = N_0 e^{kt} \]
We are given that after 8 hours:
\[ \ln\left(\frac{N(8)}{N_0}\right) = 4 \]
This implies:
\[ \frac{N(8)}{N_0} = e^4 \]
Thus:
\[ N(8) = N_0 e^4 \]
This equation can also be expressed as \( N(8) = N_0 e^{8k} \).
Equating both expressions for \( N(8) \):
\[ N_0 e^4 = N_0 e^{8k} \]
Solving for \( k \):
\[ 4 = 8k \]
\[ k = \frac{4}{8} = \frac{1}{2} \]
Therefore, the growth constant \( k \) is \(\frac{1}{2}\).
##### Step 3: Calculating the Ratio after 10 Hours
Substitute \( k = \frac{1}{2} \) and \( t = 10 \) hours into the standard equation:
\[ N(t) = N_0 e^{\left(\frac{1}{2} \cdot 10\right)} \]
\[ N(10) = N_0 e^5 \]
The ratio of the population to the initial population after 10 hours is:
\[ \frac{N(10)}{N_0} = e^5
![### Homemade Yogurt Recipe
1. **Heat milk above 180°F.**
2. **Cool milk and hold at 110°F.**
3. **Add starter and stir.**
4. **Incubate for 7-9 hours.**
### Research Context
Data from a 1990 research paper indicated that *Lactobacillus plantarum* had:
\[ \ln\left(\frac{N}{N_0}\right) = 4 \]
where \( t = 8 \) hours.
- \( N \) is the number of bacteria at \( t = 8 \) hours.
- \( N_0 \) is the initial number of bacteria.
### Mathematical Model
To find the growth constant \( k \) in the equation:
\[ N(t) = N_0 e^{kt} \]
Given that:
\[ \ln\left(\frac{N(8)}{N_0}\right) = 4 \]
\[ \frac{N(8)}{N_0} = e^4 \]
This implies:
\[ N(8) = N_0 e^{8k} \]
At \( N_0 e^4 = N_0 e^{8k} \), solving for \( k \):
\[ 4 = 8k \]
\[ k = \frac{4}{8} = \frac{1}{2} \]
### Explanation
- The left side includes a simple four-step process for making homemade yogurt.
- The right side explains logarithmic growth of *Lactobacillus plantarum* with a mathematical derivation to find the growth constant \( k \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F113b0878-5af8-4998-be4b-de5d66497a2a%2F12abdf4f-5125-41f5-ba14-9fad39797b4e%2Fdbewj7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Homemade Yogurt Recipe
1. **Heat milk above 180°F.**
2. **Cool milk and hold at 110°F.**
3. **Add starter and stir.**
4. **Incubate for 7-9 hours.**
### Research Context
Data from a 1990 research paper indicated that *Lactobacillus plantarum* had:
\[ \ln\left(\frac{N}{N_0}\right) = 4 \]
where \( t = 8 \) hours.
- \( N \) is the number of bacteria at \( t = 8 \) hours.
- \( N_0 \) is the initial number of bacteria.
### Mathematical Model
To find the growth constant \( k \) in the equation:
\[ N(t) = N_0 e^{kt} \]
Given that:
\[ \ln\left(\frac{N(8)}{N_0}\right) = 4 \]
\[ \frac{N(8)}{N_0} = e^4 \]
This implies:
\[ N(8) = N_0 e^{8k} \]
At \( N_0 e^4 = N_0 e^{8k} \), solving for \( k \):
\[ 4 = 8k \]
\[ k = \frac{4}{8} = \frac{1}{2} \]
### Explanation
- The left side includes a simple four-step process for making homemade yogurt.
- The right side explains logarithmic growth of *Lactobacillus plantarum* with a mathematical derivation to find the growth constant \( k \).
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