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.

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?

1. As t increases, p increases slowly at first, then quickly.
2. The y-intercept is greater than 0.
3. There is a horizontal asymptote at p = 0.
4. As t increases, p decreases at a decreasing rate.
5. The p-values include all real numbers.
6. The graph decreases without bound as t approaches positive infinity.

GPT-4o mini · Answer

Let's analyze the given equation for the rhinoceros population: \( p(t) = 60(0.92)^t \).

As t increases, p increases slowly at first, then quickly.
- False. The population is modeled by an exponential decay function (since the base \(0.92 < 1\)), so as \(t\) increases, \(p(t)\) decreases. It does not increase at all.
The y-intercept is greater than 0.
- True. The y-intercept is found by evaluating \(p(0)\): \( p(0) = 60(0.92)^0 = 60 \), which is indeed greater than 0.
There is a horizontal asymptote at p = 0.
- True. Since the population decreases exponentially and is always positive, as \(t\) approaches infinity, \(p(t)\) approaches 0 but never actually reaches it, creating a horizontal asymptote at \(p = 0\).
As t increases, p decreases at a decreasing rate.
- False. The decrease in \(p(t)\) is exponential, which means it decreases at an increasing rate (a constant percentage decrease each year) rather than a decreasing rate.
The p-values include all real numbers.
- False. The p-values (the population) are always positive, so they cannot include all real numbers. They are in the range \( (0, 60] \).
The graph decreases without bound as t approaches positive infinity.
- True. As \(t\) increases, \(p(t)\) gets closer and closer to 0 but never actually becomes negative, meaning it decreases toward 0 without bound.

Summary of True Statements

2. True
3. True
6. True

Therefore, the true statements are 2, 3, and 6.