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

(1 point)
Responses

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).
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 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 .

Answer 1

Let's analyze the equation \( m(p) = 20(2^p) \).

1. Understanding the Equation:

   • \( 20 \) is the initial amount of money.
   • \( 2^p \) indicates that for each person who passes it on, the amount of money doubles.
   • As \( p \) (the number of people who pass the money on) increases, \( m \) (the total amount of money) increases exponentially.

2. Behavior of the Function:

   • As \( p \) increases, \( m \) increases quickly (exponentially), meaning the amount of money grows rapidly as more people opt to pass it on.
   • Thus, the statement "As p increases, m increases quickly" is true.

3. Domain of the Function:

   • Here, \( p \) can be any non-negative integer (0, 1, 2, ...), meaning \( p \) can take values greater than or equal to 0, so the domain is \( p \geq 0 \) and not \( p \geq 20 \).

4. Range of the Function:

   • The minimum amount when \( p = 0 \) is \( m(0) = 20(2^0) = 20 \).
   • As \( p \) increases, \( m \) grows without bound. Therefore, the range of \( m \) is \( m \geq 20 \).

Based on this analysis, we can analyze the provided statements:

1. "As p increases, m increases quickly at first and then slowly." – This is not true; it increases quickly and continues to do so.
2. "The domain (p-values) include the values greater than or equal to 20 (p≥20)." – This is not true; the domain starts from 0, i.e., \( p \geq 0 \).
3. "As p increases, m decreases slowly at first and then quickly." – This is not true; \( m \) does not decrease.
4. "The range (m-values) includes the values (m≥20)." – This is true; the minimum value is 20, and it increases without bound.

Thus, the correct answer is:

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