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The rhinoceros is one of the world’s most endangered animals. One specific species has diminished to approximately 60 rhinocero...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
A: The true statements are 2, 3, 4, and 6.
B: The true statements are 2, 4, 5, and 6.
C: The true statements are 2, 4, and 5.
D: The true statements are 1, 2, 3, and 6.
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
A: The true statements are 2, 3, 4, and 6.
B: The true statements are 2, 4, 5, and 6.
C: The true statements are 2, 4, and 5.
D: The true statements are 1, 2, 3, and 6.
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To evaluate the statements based on the population model \( p(t) = 60(0.92)^t \), let’s analyze each statement:
1. **Statement #1**: As t increases, p increases slowly at first, then quickly.
- **False**: The population is decreasing, not increasing.
2. **Statement #2**: The y-intercept is greater than 0.
- **True**: When \( t = 0 \), \( p(0) = 60(0.92)^0 = 60 \). The y-intercept (population at year 0) is 60, which is greater than 0.
3. **Statement #3**: There is a horizontal asymptote at p = 0.
- **True**: As \( t \) approaches infinity \( (t \to \infty) \), \( p(t) \) approaches 0, indicating a horizontal asymptote at \( p = 0 \).
4. **Statement #4**: As t increases, p decreases at a decreasing rate.
- **True**: The population decreases by a constant percentage (8% per year), which means it decreases at a decreasing rate rather than a constant amount.
5. **Statement #5**: The p-values include all real numbers.
- **False**: The population \( p(t) \) can never be negative or exceed 60, so the p-values are restricted to the interval \( (0, 60] \).
6. **Statement #6**: The graph decreases without bound as t approaches positive infinity.
- **True**: While the population approaches 0, it does decrease continuously but does not go below 0. It "approaches" 0 without actually reaching it.
Now, let's summarize the true statements:
- True Statements: 2, 3, 4, and 6
- False Statements: 1 and 5
Given this analysis, the correct answer is:
**A: The true statements are 2, 3, 4, and 6.**
1. **Statement #1**: As t increases, p increases slowly at first, then quickly.
- **False**: The population is decreasing, not increasing.
2. **Statement #2**: The y-intercept is greater than 0.
- **True**: When \( t = 0 \), \( p(0) = 60(0.92)^0 = 60 \). The y-intercept (population at year 0) is 60, which is greater than 0.
3. **Statement #3**: There is a horizontal asymptote at p = 0.
- **True**: As \( t \) approaches infinity \( (t \to \infty) \), \( p(t) \) approaches 0, indicating a horizontal asymptote at \( p = 0 \).
4. **Statement #4**: As t increases, p decreases at a decreasing rate.
- **True**: The population decreases by a constant percentage (8% per year), which means it decreases at a decreasing rate rather than a constant amount.
5. **Statement #5**: The p-values include all real numbers.
- **False**: The population \( p(t) \) can never be negative or exceed 60, so the p-values are restricted to the interval \( (0, 60] \).
6. **Statement #6**: The graph decreases without bound as t approaches positive infinity.
- **True**: While the population approaches 0, it does decrease continuously but does not go below 0. It "approaches" 0 without actually reaching it.
Now, let's summarize the true statements:
- True Statements: 2, 3, 4, and 6
- False Statements: 1 and 5
Given this analysis, the correct answer is:
**A: The true statements are 2, 3, 4, and 6.**
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