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

, 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 .

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

The domain (p-values) include the values greater than or equal to 20 (p≥20)
.

Answer 1

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

1. Understanding the equation:

   • The equation represents the amount of money $m$ as a function of the number of people $p$ who choose to pass on the money.
   • Initially, when $p=0$, $m(0)=20(2^0)=20$.
   • As $p$ increases, the term $2^p$ grows exponentially, meaning that $m(p)$ grows very quickly.

2. Analysis of options:

   • "As p increases, m decreases slowly at first and then quickly." - This is incorrect because $m$ actually increases as $p$ increases.
   • "The range (m-values) includes the values (m ≥ 20)." - This is correct. The minimum value of $m$ occurs when $p=0$, and $m(0)=20$. As $p$ increases, $m$ continues to increase.
   • "As p increases, m increases quickly at first and then slowly." - This is incorrect. The nature of exponential growth means $m$ increases slowly for small $p$ and then increases rapidly as $p$ increases further.
   • "The domain (p-values) includes the values greater than or equal to 20 (p ≥ 20)." - This is incorrect. $p$ can take non-negative integer values starting from 0, which means $p$ can be 0, 1, 2, 3, etc.

Based on this analysis, the correct statement is:

The range (m-values) includes the values (m ≥ 20).