### Red-Eared Slider Turtle Population StudyMarjorie Lee Browne is studying the population of red-eared slider turtles and has developed the following equation to represent the turtle population $ t $ years after she initially started studying them:\[ p(t) = 12(1.4^t) \]#### Questions and Analysis1. **Is this an exponential growth or decay function?** - A. Growth; $ b > 1 $ - B. Growth; $ a > 1 $ - C. Growth; $ 0 < b < 1 $ - D. Decay; $ b > 1 $ - E. Decay; $ 0 < a < 1 $ - F. Decay; $ 0 < b < 1 $2. **What is the vertical intercept?** - Write your answer as a point.3. **What are the units for the nonzero coordinate of the vertical intercept?** - A. days - B. years - C. turtles4. **What is the growth/decay rate for the turtle population?** - Write your answer as a percentage.5. **What is the turtle population four years after she has started studying them?** - Round to the nearest whole number.6. **At the start of which year will the turtle population reach 100?**### Explanation#### Function TypeThe function $ p(t) = 12(1.4^t) $ represents an exponential growth because the base of the exponent (1.4) is greater than 1.#### Vertical InterceptThe vertical intercept occurs at $ t = 0 $. - Calculation: $ p(0) = 12(1.4^0) = 12 $- Vertical Intercept: $(0, 12)$#### Units for Vertical InterceptThe units for the nonzero coordinate of the vertical intercept are turtles.#### Growth RateThe growth rate can be found by subtracting 1 from the base and converting to a percentage: - Growth rate: $(1.4 - 1) \times 100\% = 40\%$#### Turtle Population After Four YearsCalculate using the equation:- $ p(4) = 12(1.4^4) **Problem Context and Graph Analysis****b. What does the growth/decay factor mean in the context of the problem?***This space is designated for an explanation of the growth or decay factor within the specific context provided in the problem. Learners should think about how the factor reflects changes over time in the given scenario.***c. Which of the following is the graph of \( p(t) $?**The task is to identify the correct graph representation for the function $ p(t) $. Below are descriptions of the four graph options:1. **Graph 1**: Displays exponential growth. The curve starts off slowly increasing and accelerates significantly as it moves to the right. 2. **Graph 2**: Shows exponential decay. The curve rapidly decreases and then levels out as it approaches the x-axis. 3. **Graph 3**: Similar to Graph 1, exhibiting exponential growth with an upward curve, starting off slowly and then steepening.4. **Graph 4**: Exhibits a more linear progression, but curves slightly upwards, suggesting potential growth, though not as pronounced as in Graphs 1 and 3.Students should evaluate which graph accurately reflects the behavior of $ p(t) $ based on its mathematical characteristics.

Answered: Image /qna-images/answer/7608489b-6ad0-4e37-a9dd-36e124dc45ef.jpg

Marjorie Lee Browne is studying the population of red-eared slider turtles and has come up with the following equation to represent the turtle populationt years after she initially started studying them. p(t) = 12(1.4") %3D Is this an exponential growth or decay function? O A. Growth; b>1 O B. Growth; a > 1 OC. Growth; 0 < b<1 OD. Decay; b >1 OE. Decay; 0

Marjorie Lee Browne is studying the population of red-eared slider turtles and has come up with the following equation to represent the turtle populationt years after she initially started studying them. p(t) = 12(1.4") %3D Is this an exponential growth or decay function? O A. Growth; b>1 O B. Growth; a > 1 OC. Growth; 0 < b<1 OD. Decay; b >1 OE. Decay; 0

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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### Red-Eared Slider Turtle Population Study

Marjorie Lee Browne is studying the population of red-eared slider turtles and has developed the following equation to represent the turtle population $ t $ years after she initially started studying them:

\[ p(t) = 12(1.4^t) \]

#### Questions and Analysis

1. **Is this an exponential growth or decay function?**
- A. Growth; $ b > 1 $
- B. Growth; $ a > 1 $
- C. Growth; $ 0 < b < 1 $
- D. Decay; $ b > 1 $
- E. Decay; $ 0 < a < 1 $
- F. Decay; $ 0 < b < 1 $

2. **What is the vertical intercept?**
- Write your answer as a point.

3. **What are the units for the nonzero coordinate of the vertical intercept?**
- A. days
- B. years
- C. turtles

4. **What is the growth/decay rate for the turtle population?**
- Write your answer as a percentage.

5. **What is the turtle population four years after she has started studying them?**
- Round to the nearest whole number.

6. **At the start of which year will the turtle population reach 100?**

### Explanation

#### Function Type
The function $ p(t) = 12(1.4^t) $ represents an exponential growth because the base of the exponent (1.4) is greater than 1.

#### Vertical Intercept
The vertical intercept occurs at $ t = 0 $.
- Calculation: $ p(0) = 12(1.4^0) = 12 $
- Vertical Intercept: $(0, 12)$

#### Units for Vertical Intercept
The units for the nonzero coordinate of the vertical intercept are turtles.

#### Growth Rate
The growth rate can be found by subtracting 1 from the base and converting to a percentage:
- Growth rate: $(1.4 - 1) \times 100\% = 40\%$

#### Turtle Population After Four Years
Calculate using the equation:
- \( p(4) = 12(1.4^4)

**Problem Context and Graph Analysis**

**b. What does the growth/decay factor mean in the context of the problem?**

*This space is designated for an explanation of the growth or decay factor within the specific context provided in the problem. Learners should think about how the factor reflects changes over time in the given scenario.*

**c. Which of the following is the graph of $ p(t) $?**

The task is to identify the correct graph representation for the function $ p(t) $. Below are descriptions of the four graph options:

1. **Graph 1**: Displays exponential growth. The curve starts off slowly increasing and accelerates significantly as it moves to the right.

2. **Graph 2**: Shows exponential decay. The curve rapidly decreases and then levels out as it approaches the x-axis.

3. **Graph 3**: Similar to Graph 1, exhibiting exponential growth with an upward curve, starting off slowly and then steepening.

4. **Graph 4**: Exhibits a more linear progression, but curves slightly upwards, suggesting potential growth, though not as pronounced as in Graphs 1 and 3.

Students should evaluate which graph accurately reflects the behavior of $ p(t) $ based on its mathematical characteristics.

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