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?

Question

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

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 .

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 .

As p increases, m decreases slowly at first and then quickly.
As p increases, m decreases slowly at first and then quickly.

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

GPT-4o mini · Answer

Let's analyze the given exponential equation $ m(p) = 20(2^p) $.

Domain (p-values):
- The variable $ p $ represents the number of people who opt to pass the money on. In this scenario, since $ p $ counts the number of exchanges and must be a non-negative integer, the domain includes values such that $ p \geq 0 $ (not 20). Hence, the correct statement about the domain would be that the domain includes non-negative integers: $ p \geq 0 $.
Range (m-values):
- We see that $ m(p) = 20(2^p) $. Since $ 2^p $ is always positive for any value of $ p $ in its domain (including $ p = 0 $ which gives $ m(0) = 20 $), the smallest value $ m $ can take is 20 when $ p=0 $, and it increases indefinitely as $ p $ increases. Thus, the range of $ m $ is $ m \geq 20 $.
Behavior of m as p increases:
- As $ p $ increases, $ 2^p $ increases exponentially. Therefore, $ m(p) = 20(2^p) $ also increases exponentially. This means that $ m $ increases quickly as $ p $ increases. So, the statement “As p increases, m increases quickly at first and then slowly” is incorrect — $ m $ does not increase slowly after initially accelerating; it continues to increase quickly.

Based on this analysis:

The correct statements are: The range (m-values) includes the values $ m \geq 20 $.

The other statements about domain and the behavior of m as p increases are incorrect.