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?

Question

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 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.

The true statements are 1, 2, 3, and 6.

Answer 1

Let's analyze each statement based on the population model \( p(t) = 60(0.92)^t \):

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

   • False: The function models a decreasing population, so as \( t \) increases, \( p(t) \) actually decreases, and it does so at a decreasing rate rather than increasing.

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

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

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

   • True: As \( t \) approaches infinity, \( p(t) \) approaches 0, so there is a horizontal asymptote at \( p = 0 \).

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

   • True: Since the population is modeled with the function \( p(t) = 60(0.92)^t \), the derivative \( p'(t) \) will show that the rate of decrease is less as \( t \) increases, indicating a decreasing rate of decrease.

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

   • False: The population \( p(t) \) cannot be negative; it only takes non-negative values.

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

   • False: While the population decreases, it approaches the asymptote at \( p=0 \) but never actually becomes negative.

Thus, the true statements are 2, 3, and 4.

Therefore, the correct response is:

The true statements are 2, 3, and 4.