Mason is completing a study for his psychology course. For the study, he begins with $20 and asks individuals on the street if they want the money or if they would like him to double it and give it to the next person. He continues to do this until someone takes the money. This situation can be modeled by the exponential equation m(p)=20(2)p , where m is the money and p is the number of people who opt to pass it on to the next person. Which of the following statements is true about the equation? (1 point) Responses As p increases, m decreases slowly at first and then quickly. As p increases, m decreases slowly at first and then quickly. The range (m-values) includes the values (m≥20). The range ( m -values) includes the values left parenthesis m greater than or equal to 20 right parenthesis . The domain (p-values) include the values greater than or equal to 20 (p≥20). The domain ( p -values) include the values greater than or equal to 20 left parenthesis p greater than or equal to 20 right parenthesis . As p increases, m increases quickly at first and then slowly.

Answer 1

The correct statement about the equation \( m(p) = 20(2^p) \) is:

As \( p \) increases, \( m \) increases quickly at first and then slowly.

Here's the reasoning:

1. Understanding the function: The equation represents an exponential function where \( m \) (the money) increases as \( p \) (the number of people passing the money) increases. Specifically, the function doubles for each additional person who passes it, starting from $20.

2. Behavior of exponential growth: In exponential functions, the growth starts gradually but accelerates as \( p \) increases. So, as more people opt to pass the money, \( m \) increases very rapidly.

3. Range: The range of the function \( m(p) \) includes values greater than or equal to $20 since the minimum amount \( m \) could be (when \( p = 0 \)) is $20, and it only gets larger as \( p \) increases.

4. Domain: The domain of \( p \) includes non-negative integers ( \( p \geq 0 \) ). The initial statement regarding \( p \) being greater than or equal to 20 is incorrect.

Thus, the only statement that is true is the one about \( m \) increasing quickly at first and then slowing down relative to \( p \).