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.