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

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
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)
(
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)
(
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.

GPT-4o mini · Answer

To analyze the equation $ m(p) = 20(2)^p $: 1. **Domain**: The variable $ p $ represents the number of people who opt to pass the money on. Since you cannot have a negative number of people passing it on, the domain of $ p $ is non-negative integers: $ p \geq 0 $. 2. **Range**: Given $ m(p) = 20(2)^p $: - When $ p = 0 $: $ m(0) = 20(2)^0 = 20 $. - As $ p $ increases, $ (2)^p $ grows rapidly, so $ m(p) $ will also increase without bound. Therefore, the range of $ m $ is $ m \geq 20 $ since the minimum value occurs when $ p = 0 $. 3. **Behavior of $ m $**: As $ p $ increases, $ m $ increases exponentially. Initially, for small values of $ p $, it might not seem to increase rapidly, but it accelerates quickly due to the exponential nature of $ (2)^p $. Reviewing your statements: - The domain includes values greater than or equal to 20 ($ p \geq 20 $): **Incorrect**. The domain is $ p \geq 0 $. - The range includes values $ m \geq 20 $: **Correct**. The range starts at 20 when $ p = 0 $ and increases indefinitely. - As $ p $ increases, $ m $ decreases slowly at first and then quickly: **Incorrect**. $ m $ increases exponentially as $ p $ increases. Thus, the correct statement about the equation is about the range, which includes the values $ m \geq 20 $.