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

Question

, 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, 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.

The true statements are 2, 3, 4, and 6.

Answer 1

To analyze the statements one by one based on the given model \( p(t) = 60(0.92)^t \), we can clarify the behavior of this exponential decay function.

1. Statement #1: As \( t \) increases, \( p \) increases slowly at first, then quickly.

   • False: This function represents population decay; it decreases rather than increases.

2. Statement #2: The y-intercept is greater than 0.

   • True: At \( t = 0 \), \( p(0) = 60(0.92)^0 = 60 \), which is indeed greater than 0.

3. Statement #3: There is a horizontal asymptote at \( p = 0 \).

   • True: As \( t \) approaches infinity, \( p(t) \) approaches 0, behaving asymptotically.

4. Statement #4: As \( t \) increases, \( p \) decreases at a decreasing rate.

   • True: The exponential decay function decreases, but the rate of decrease slows down over time.

5. Statement #5: The \( p \)-values include all real numbers.

   • False: The population \( p(t) \) will always be positive, so the \( p \)-values only include positive real numbers.

6. Statement #6: The graph decreases without bound as \( t \) approaches positive infinity.

   • True: The population continues to decrease and approaches zero but never actually reaches a negative value.

Based on this analysis, the true statements are 2, 3, 4, and 6.

The correct response is: The true statements are 2, 3, 4, and 6.