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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![### Problem 2: Rabbit Population Dynamics
The population of a certain breed of rabbits introduced onto an isolated island is modeled by the function:
\[ P(t) = \frac{63}{9 - t}, \quad (0 \le t < 9) \]
where \( t \) is measured in months.
#### Questions:
**(a)** Find the number of rabbits present on the island initially. The answer is 7.
**(b)** Show that the number of rabbits is increasing without bound.
#### Additional Questions:
**3. What procedure did you take to answer question (a)? Write the answer in a complete sentence (making sure to reference both a time and a population), using labels where appropriate.**
**4. What procedure did you take to answer question (b)? How do you know that’s the appropriate procedure?**
---
### Solution:
**(a) Initial Rabbit Population:**
To find the number of rabbits initially, we substitute \( t = 0 \) into the given function \( P(t) \):
\[ P(0) = \frac{63}{9 - 0} = \frac{63}{9} = 7 \]
Therefore, the initial number of rabbits present on the island is 7.
**(b) Increasing Without Bound:**
To show that the number of rabbits is increasing without bound, we analyze the behavior of the function \( P(t) \) as \( t \) approaches 9. Since the denominator \( 9 - t \) approaches 0 as \( t \) approaches 9 from the left, the value of \( P(t) \) increases without bound (approaches infinity).
Thus, the number of rabbits increases without bound as time approaches 9 months.
---
### Methodology for Answering:
**3. Procedure for Part (a):**
To determine the initial population of rabbits on the island, I evaluated the population function \( P(t) \) at \( t = 0 \). By substituting \( t = 0 \) into the function \( P(t) = \frac{63}{9 - t} \), I calculated the initial number of rabbits to be 7.
**4. Procedure for Part (b):**
To demonstrate that the number of rabbits is increasing without bound, I examined the behavior of the function \( P(t) = \frac{63}{9 - t} \) as \( t \) approaches 9.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77461b06-0772-4165-878f-b81200c01932%2F1d909e21-0fd7-46c3-8687-133a7dea8ed3%2Fj81cq45.png&w=3840&q=75)
Transcribed Image Text:### Problem 2: Rabbit Population Dynamics
The population of a certain breed of rabbits introduced onto an isolated island is modeled by the function:
\[ P(t) = \frac{63}{9 - t}, \quad (0 \le t < 9) \]
where \( t \) is measured in months.
#### Questions:
**(a)** Find the number of rabbits present on the island initially. The answer is 7.
**(b)** Show that the number of rabbits is increasing without bound.
#### Additional Questions:
**3. What procedure did you take to answer question (a)? Write the answer in a complete sentence (making sure to reference both a time and a population), using labels where appropriate.**
**4. What procedure did you take to answer question (b)? How do you know that’s the appropriate procedure?**
---
### Solution:
**(a) Initial Rabbit Population:**
To find the number of rabbits initially, we substitute \( t = 0 \) into the given function \( P(t) \):
\[ P(0) = \frac{63}{9 - 0} = \frac{63}{9} = 7 \]
Therefore, the initial number of rabbits present on the island is 7.
**(b) Increasing Without Bound:**
To show that the number of rabbits is increasing without bound, we analyze the behavior of the function \( P(t) \) as \( t \) approaches 9. Since the denominator \( 9 - t \) approaches 0 as \( t \) approaches 9 from the left, the value of \( P(t) \) increases without bound (approaches infinity).
Thus, the number of rabbits increases without bound as time approaches 9 months.
---
### Methodology for Answering:
**3. Procedure for Part (a):**
To determine the initial population of rabbits on the island, I evaluated the population function \( P(t) \) at \( t = 0 \). By substituting \( t = 0 \) into the function \( P(t) = \frac{63}{9 - t} \), I calculated the initial number of rabbits to be 7.
**4. Procedure for Part (b):**
To demonstrate that the number of rabbits is increasing without bound, I examined the behavior of the function \( P(t) = \frac{63}{9 - t} \) as \( t \) approaches 9.
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