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.

1 answer

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

  1. 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.
  2. 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.
  3. 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 \(p = 0\).
  4. 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.
  5. 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 \( (0, 60] \).
  6. 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.