Let's analyze each statement about the model \( p(t) = 60(0.92)^t \):
Statement #1: As t increases, p increases slowly at first, then quickly.
- False. The given model represents a decreasing exponential function. As \( t \) increases, \( p(t) \) decreases, not increases.
Statement #2: The y-intercept is greater than 0.
- True. The y-intercept occurs at \( t = 0 \). Plugging in \( t = 0 \) gives \( p(0) = 60(0.92)^0 = 60 \), which is greater than 0.
Statement #3: There is a horizontal asymptote at p = 0.
- True. Since the population is decreasing exponentially, as time goes to infinity (\( t \to \infty \)), \( p(t) \) approaches 0, indicating a horizontal asymptote at \( p = 0 \).
Statement #4: As t increases, p decreases at a decreasing rate.
- False. While \( p(t) \) does decrease over time, it actually decreases at a constant percentage rate (8%). Therefore, the absolute rate of change continues to decrease, but the rate itself decreases; it decreases continuously.
Statement #5: The p-values include all real numbers.
- False. The population \( p(t) \) is always positive and approaches 0, but cannot be negative. Therefore, the p-values do not include all real numbers.
Statement #6: The graph decreases without bound as t approaches positive infinity.
- True. As \( t \) goes to infinity, \( p(t) \) approaches 0, which means the graph decreases without bound, but not below 0.
In summary, the true statements are:
- Statement #2: True
- Statement #3: True
- Statement #6: True
The false statements are:
- Statement #1: False
- Statement #4: False
- Statement #5: False