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

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.

Answer 1

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.