Let's analyze the provided statements about the population growth of the Louisiana black bear modeled by the equation \( p(t) = 150(1.053)^t \).
Statement #1: As t increases, p increases slowly at first, then quickly.
True. This is characteristic of exponential growth. In the initial phases, the growth rate appears linear (growing slowly), but as time goes on, the effect of compounding leads to a much faster increase in population.
Statement #2: The y-intercept is greater than 0.
True. The y-intercept occurs when \( t = 0 \). Substituting into the equation: \[ p(0) = 150(1.053)^0 = 150(1) = 150 \] Thus, the y-intercept is 150, which is greater than 0.
Statement #3: There is a vertical asymptote at p=0.
False. While the function approaches zero as \( t \) approaches negative infinity, exponential growth functions do not possess vertical asymptotes in the same way that rational functions do. This equation represents a population that will increase indefinitely as \( t \) increases, but it does not have a vertical asymptote at \( p=0 \).
Statement #4: As t increases, p increases at a decreasing rate.
False. As described in statement #1, while the initial growth may seem slow, the rate of increase does not decrease; rather, it increases at an increasing rate due to the properties of exponential functions. The growth is compounding, meaning the increases become larger over time.
Summary of the validity of statements:
- Statement #1: True
- Statement #2: True
- Statement #3: False
- Statement #4: False
Thus, the statements about the graph of the Louisiana black bear population growth are correctly identified, with only statements #1 and #2 being true.
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