Question
Question
The rhinoceros is one of the world’s most endangered animals. One specific species has diminished to approximately 60 rhinoceroses, and the population is decreasing at a rate of 8 percent per year. The situation can be modeled by the equation p(t)=60(0.92)t
, where p is the population of rhinoceroses after t years. Which of the following statements about the key features of the exponential graph for this situation are true?
Statement #1: As t increases, p increases slowly at first, then quickly.
Statement #2: The y-intercept is greater than 0.
Statement #3: There is a horizontal asymptote at p=0
.
Statement #4: As t increases, p decreases at a decreasing rate.
Statement #5: The p-values include all real numbers.
Statement #6: The graph decreases without bound as t approaches positive infinity.
(1 point)
Responses
The true statements are 1, 2, 3, and 6.
The true statements are 1, 2, 3, and 6.
The true statements are 2, 3, 4, and 6.
The true statements are 2, 3, 4, and 6.
The true statements are 2, 4, 5, and 6.
The true statements are 2, 4, 5, and 6.
The true statements are 2, 4, and 5.
The true statements are 2, 4, and 5.
Skip to navigation
The rhinoceros is one of the world’s most endangered animals. One specific species has diminished to approximately 60 rhinoceroses, and the population is decreasing at a rate of 8 percent per year. The situation can be modeled by the equation p(t)=60(0.92)t
, where p is the population of rhinoceroses after t years. Which of the following statements about the key features of the exponential graph for this situation are true?
Statement #1: As t increases, p increases slowly at first, then quickly.
Statement #2: The y-intercept is greater than 0.
Statement #3: There is a horizontal asymptote at p=0
.
Statement #4: As t increases, p decreases at a decreasing rate.
Statement #5: The p-values include all real numbers.
Statement #6: The graph decreases without bound as t approaches positive infinity.
(1 point)
Responses
The true statements are 1, 2, 3, and 6.
The true statements are 1, 2, 3, and 6.
The true statements are 2, 3, 4, and 6.
The true statements are 2, 3, 4, and 6.
The true statements are 2, 4, 5, and 6.
The true statements are 2, 4, 5, and 6.
The true statements are 2, 4, and 5.
The true statements are 2, 4, and 5.
Skip to navigation
Answers
GPT-4o mini
Let's analyze the statements one by one regarding the population function \( p(t) = 60(0.92)^t \):
1. **Statement #1: As t increases, p increases slowly at first, then quickly.**
**False.** The function represents exponential decay since the base (0.92) is less than 1. As \( t \) increases, \( p(t) \) decreases.
2. **Statement #2: The y-intercept is greater than 0.**
**True.** When \( t = 0 \), \( p(0) = 60(0.92)^0 = 60 > 0 \).
3. **Statement #3: There is a horizontal asymptote at p = 0.**
**True.** As \( t \) approaches infinity, \( p(t) \) approaches 0, giving us a horizontal asymptote at \( p = 0 \).
4. **Statement #4: As t increases, p decreases at a decreasing rate.**
**True.** The rate of decrease becomes smaller over time in an exponential decay model.
5. **Statement #5: The p-values include all real numbers.**
**False.** The population values \( p \) are limited to non-negative numbers (it cannot be less than 0).
6. **Statement #6: The graph decreases without bound as t approaches positive infinity.**
**True.** As \( t \) approaches infinity, \( p(t) \) gets closer to 0 but does not become negative.
### Summary of Truth Value for Each Statement:
- Statement #1: False
- Statement #2: True
- Statement #3: True
- Statement #4: True
- Statement #5: False
- Statement #6: True
### True Statements:
The true statements are 2, 3, 4, and 6.
Therefore, the correct response is:
**The true statements are 2, 3, 4, and 6.**
1. **Statement #1: As t increases, p increases slowly at first, then quickly.**
**False.** The function represents exponential decay since the base (0.92) is less than 1. As \( t \) increases, \( p(t) \) decreases.
2. **Statement #2: The y-intercept is greater than 0.**
**True.** When \( t = 0 \), \( p(0) = 60(0.92)^0 = 60 > 0 \).
3. **Statement #3: There is a horizontal asymptote at p = 0.**
**True.** As \( t \) approaches infinity, \( p(t) \) approaches 0, giving us a horizontal asymptote at \( p = 0 \).
4. **Statement #4: As t increases, p decreases at a decreasing rate.**
**True.** The rate of decrease becomes smaller over time in an exponential decay model.
5. **Statement #5: The p-values include all real numbers.**
**False.** The population values \( p \) are limited to non-negative numbers (it cannot be less than 0).
6. **Statement #6: The graph decreases without bound as t approaches positive infinity.**
**True.** As \( t \) approaches infinity, \( p(t) \) gets closer to 0 but does not become negative.
### Summary of Truth Value for Each Statement:
- Statement #1: False
- Statement #2: True
- Statement #3: True
- Statement #4: True
- Statement #5: False
- Statement #6: True
### True Statements:
The true statements are 2, 3, 4, and 6.
Therefore, the correct response is:
**The true statements are 2, 3, 4, and 6.**