t = years since 1940 N = cubic ft. in trillions 3.75 8.48 10 15.09 23.79 20 30 40 21.87 50 21.52 60 24.15 (a) Make a logistic model for N as a function of t using Desmos. 12.54 O N(t) = 1+ 5.94e-0.35t 35.67 O N(t) = 1+ 3.41e-1.25t 23.31 O N(t) = 1+ 6.99e¬0.14' 34.86 O N(t) = 1+ 8.54e-0.912 29.53 O N(t) = 1+ 7.58e-0.88? (b) Graph the data and the logistic model.
t = years since 1940 N = cubic ft. in trillions 3.75 8.48 10 15.09 23.79 20 30 40 21.87 50 21.52 60 24.15 (a) Make a logistic model for N as a function of t using Desmos. 12.54 O N(t) = 1+ 5.94e-0.35t 35.67 O N(t) = 1+ 3.41e-1.25t 23.31 O N(t) = 1+ 6.99e¬0.14' 34.86 O N(t) = 1+ 8.54e-0.912 29.53 O N(t) = 1+ 7.58e-0.88? (b) Graph the data and the logistic model.
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
![**Transcription:**
**Graphs Explanation:**
The image contains four graphs arranged in two rows and two columns. Each graph depicts a logistic growth model with actual data points overlaid.
- **Axes:**
- The x-axis is labeled as "t," likely representing time.
- The y-axis is labeled as "N," representing the quantity or size of the population/resource.
- **Graphs Description:**
- A solid curve represents the logistic growth model.
- Discrete data points (marked by dots) are plotted over this curve.
- The graphs demonstrate how well the model fits the actual data at various stages.
- Each graph has a small circle at the bottom left, possibly indicating a selection or annotation option.
**Questions:**
**(c)** Which year's production was farthest from the prediction of the logistic model?
[ ]
**(d)** What does the logistic model predict for the amount of natural gas that will be produced in the long run? *Note: In other contexts, this would be known as the carrying capacity. (Use the model found in part (a).)*
[ ] trillion](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3455011a-16e4-48f4-b0aa-48b6e4cda83a%2F592e91a6-3fa2-4125-b8f1-024df6e77166%2Fp7o30x_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription:**
**Graphs Explanation:**
The image contains four graphs arranged in two rows and two columns. Each graph depicts a logistic growth model with actual data points overlaid.
- **Axes:**
- The x-axis is labeled as "t," likely representing time.
- The y-axis is labeled as "N," representing the quantity or size of the population/resource.
- **Graphs Description:**
- A solid curve represents the logistic growth model.
- Discrete data points (marked by dots) are plotted over this curve.
- The graphs demonstrate how well the model fits the actual data at various stages.
- Each graph has a small circle at the bottom left, possibly indicating a selection or annotation option.
**Questions:**
**(c)** Which year's production was farthest from the prediction of the logistic model?
[ ]
**(d)** What does the logistic model predict for the amount of natural gas that will be produced in the long run? *Note: In other contexts, this would be known as the carrying capacity. (Use the model found in part (a).)*
[ ] trillion
Transcribed Image Text:### Logistic Growth Model Analysis
The table below lists data for cubic feet in trillions of a substance over time:
| \( t \) (years since 1940) | \( N \) (cubic ft. in trillions) |
|----------------------------|---------------------------------|
| 0 | 3.75 |
| 10 | 8.48 |
| 20 | 15.09 |
| 30 | 23.79 |
| 40 | 21.87 |
| 50 | 21.52 |
| 60 | 24.15 |
#### (a) Logistic Model for \( N \) as a Function of \( t \)
Choose the appropriate logistic model using Desmos:
1. \( N(t) = \frac{12.54}{1 + 5.94e^{-0.35t}} \)
2. \( N(t) = \frac{35.67}{1 + 3.41e^{-1.25t}} \)
3. \( N(t) = \frac{23.31}{1 + 6.99e^{-0.14t}} \)
4. \( N(t) = \frac{34.86}{1 + 8.54e^{-0.91t}} \)
5. \( N(t) = \frac{29.53}{1 + 7.58e^{-0.88t}} \)
#### (b) Graph the Data and Logistic Model
Two graphs are displayed:
- **Left Graph:** This graph plots the given data points and overlays a logistic curve representing the chosen logistic growth model. On the x-axis is time (\( t \)) and on the y-axis is the quantity (\( N \)) in trillion cubic feet.
- **Right Graph:** Similar to the left graph, it plots the same data with a slightly varied logistic model curve.
The logistic curve illustrates how the quantity \( N \) grows over time, indicating a rapid growth phase followed by a plateau. The data points are used to fit the logistic model accurately.
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 4 steps with 3 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,
