The Louisiana black bear was once on the endangered species list. It was thought that as few as 150 bears remained in the wild. Because of conservationists, the population has been increasing. Assuming the population has increased at a rate of approximately 5.3

Question

percent over the last 30 years, the situation can be modeled by the equation p(t)=150(1.053)t
, where p is the population of the Louisiana black bear after t years. Which of the following statements about the key features of the exponential graph for this situation are true?

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 vertical asymptote at p=0
.

Statement #4: As t increases, p increases at a decreasing rate.

(1 point)
Responses

The true statements are 2 and 4.
The true statements are 2 and 4.

The true statements are 1 and 3.
The true statements are 1 and 3.

The true statements are 1 and 2.
The true statements are 1 and 2.

The true statements are 3 and 4.
The true statements are 3 and 4.
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Answer 1

Let's analyze each statement regarding the function \( p(t) = 150(1.053)^t \) that models the population of the Louisiana black bear:

Statement #1: As t increases, p increases slowly at first, then quickly.

• This statement is true. Exponential growth starts off slowly and then accelerates as time increases due to the nature of exponential functions.

Statement #2: The y-intercept is greater than 0.

• The y-intercept occurs when \( t = 0 \). Plugging in \( t = 0 \) gives us \( p(0) = 150(1.053)^0 = 150 \). Therefore, the y-intercept is 150, which is greater than 0. This statement is true.

Statement #3: There is a vertical asymptote at p=0.

• In the context of this exponential function, there is no vertical asymptote. Instead, as \( t \) approaches infinity, \( p(t) \) approaches infinity. Thus, this statement is false.

Statement #4: As t increases, p increases at a decreasing rate.

• This statement is false. In exponential growth, while the increase in population might appear less significant over smaller intervals as \( t \) increases, it never truly decreases. The rate of increase is proportionally larger as \( p(t) \) grows, so the amount added is increasing, not decreasing.

Based on the analysis above, the true statements are 1 and 2! Therefore, the correct answer is:

The true statements are 1 and 2.