6. The bacteria population in Colony A grows according to the recursive formula. The table shows the bacteria population of Colony B at different points in time. The initial population for Colony B is less than the population of Colony A. Will Colony B always be the smaller colony? Justify your reasoning. (F.IF.9) Time S a₁ a₁ = 2000 an = 1.1an-1 Colony B Initial 1000 After 1 hour 1200 After 2 hours 1440 After 3 hours 1728
6. The bacteria population in Colony A grows according to the recursive formula. The table shows the bacteria population of Colony B at different points in time. The initial population for Colony B is less than the population of Colony A. Will Colony B always be the smaller colony? Justify your reasoning. (F.IF.9) Time S a₁ a₁ = 2000 an = 1.1an-1 Colony B Initial 1000 After 1 hour 1200 After 2 hours 1440 After 3 hours 1728
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
help, thank you .
![### Bacteria Population Dynamics in Colony A and B
**Problem Statement:**
The bacteria population in Colony A grows according to the recursive formula given below. The table shows the bacteria population of Colony B at different points in time. The initial population for Colony B is less than the population of Colony A. The question to be addressed is: Will Colony B always be the smaller colony? Please justify your reasoning.
#### Recursive Formula for Colony A:
\[
\begin{cases}
a_1 = 2000 \\
a_n = 1.1a_{n-1}
\end{cases}
\]
#### Bacteria Population in Colony B:
| Time | Colony B |
|---------------|----------|
| Initial | 1000 |
| After 1 hour | 1200 |
| After 2 hours | 1440 |
| After 3 hours | 1728 |
#### Analysis:
1. **Initial Population:**
- Colony A starts with 2000 bacteria.
- Colony B starts with 1000 bacteria.
2. **Population Growth Rate:**
- Colony A grows by a factor of 1.1 every hour.
- Colony B's population is given at specific time points.
3. **Calculation for Colony A:**
- After 1 hour: \( a_2 = 1.1 \times 2000 = 2200 \)
- After 2 hours: \( a_3 = 1.1 \times 2200 = 2420 \)
- After 3 hours: \( a_4 = 1.1 \times 2420 = 2662 \)
4. **Comparison:**
- At the initial time, Colony A has 2000 bacteria and Colony B has 1000 bacteria.
- After 1 hour, Colony A has 2200 bacteria and Colony B has 1200 bacteria.
- After 2 hours, Colony A has 2420 bacteria and Colony B has 1440 bacteria.
- After 3 hours, Colony A has 2662 bacteria and Colony B has 1728 bacteria.
Since Colony A consistently has a higher population than Colony B at each time point, we can conclude that Colony B will always have a smaller population than Colony A based on the given recursive formula and data set.
#### Conclusion:
Colony B will always be the smaller colony compared](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F51c63e6e-e160-49df-b51c-19d982aff1f3%2F0b6235d3-a804-4989-a013-b26ff736377d%2Fvlfjeui_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Bacteria Population Dynamics in Colony A and B
**Problem Statement:**
The bacteria population in Colony A grows according to the recursive formula given below. The table shows the bacteria population of Colony B at different points in time. The initial population for Colony B is less than the population of Colony A. The question to be addressed is: Will Colony B always be the smaller colony? Please justify your reasoning.
#### Recursive Formula for Colony A:
\[
\begin{cases}
a_1 = 2000 \\
a_n = 1.1a_{n-1}
\end{cases}
\]
#### Bacteria Population in Colony B:
| Time | Colony B |
|---------------|----------|
| Initial | 1000 |
| After 1 hour | 1200 |
| After 2 hours | 1440 |
| After 3 hours | 1728 |
#### Analysis:
1. **Initial Population:**
- Colony A starts with 2000 bacteria.
- Colony B starts with 1000 bacteria.
2. **Population Growth Rate:**
- Colony A grows by a factor of 1.1 every hour.
- Colony B's population is given at specific time points.
3. **Calculation for Colony A:**
- After 1 hour: \( a_2 = 1.1 \times 2000 = 2200 \)
- After 2 hours: \( a_3 = 1.1 \times 2200 = 2420 \)
- After 3 hours: \( a_4 = 1.1 \times 2420 = 2662 \)
4. **Comparison:**
- At the initial time, Colony A has 2000 bacteria and Colony B has 1000 bacteria.
- After 1 hour, Colony A has 2200 bacteria and Colony B has 1200 bacteria.
- After 2 hours, Colony A has 2420 bacteria and Colony B has 1440 bacteria.
- After 3 hours, Colony A has 2662 bacteria and Colony B has 1728 bacteria.
Since Colony A consistently has a higher population than Colony B at each time point, we can conclude that Colony B will always have a smaller population than Colony A based on the given recursive formula and data set.
#### Conclusion:
Colony B will always be the smaller colony compared
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
