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?

Question

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 horizontal asymptote at p=0 .

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

Statement #5: The p-values include all real numbers.

Statement #6: The graph decreases without bound as t approaches positive infinity.

Answer 1

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

Answer 2

are you sure

Answer 3

Let’s double-check the analysis for each statement related to the equation \( p(t) = 60(0.92)^t \) to ensure we have a clear understanding of the situation.

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

• False. The population \( p(t) \) is actually decreasing as \( t \) increases since \( 0.92 < 1 \). Therefore, it decreases over time, not increases.

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

• True. The y-intercept can be found by evaluating \( p(0) \): \[ p(0) = 60(0.92)^0 = 60(1) = 60 \] which is indeed greater than 0.

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

• True. As \( t \) approaches infinity, \( (0.92)^t \) approaches 0; thus \( p(t) \) approaches 0. This indicates a horizontal asymptote at \( p = 0 \).

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

• False. The function decreases at a constant rate of 8% per year, meaning the amount it decreases each year is a fixed percentage of the current value, so the absolute value of the decrease gets smaller over time, but it's not decreasing at a decreasing rate; the percentage decrease remains the same.

Statement #5: The p-values include all real numbers.

• False. The population \( p(t) \) remains positive for all \( t \) values and approaches 0, but it does not include negative values.

Statement #6: The graph decreases without bound as t approaches positive infinity.

• True. The value of \( p(t) \) approaches 0 as \( t \) approaches infinity, which means it does decrease towards 0, but it will never actually go below 0.

So the final evaluation is:

• True Statements: #2, #3, #6
• False Statements: #1, #4, #5

Thank you for your patience, and yes, the previous response stands correct after the re-evaluation. If you have any further questions or need more clarifications, feel free to ask!