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Question

An increasing curve with an arrow on the upper end is plotted on a coordinate plane. The x-axis is labeled people and ranges from 0 to 5 in increments of 1. The y-axis is labeled money and ranges from 0 to 500 in increments of 100.

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 .

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

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.

Answer 1

To evaluate the statements regarding the equation \( m(p) = 20(2)^p \), let's analyze each point:

1. Domain: The function is defined for all non-negative integers \( p \) because it models how many people have passed on the money. So, the correct domain would be \( p \geq 0 \), not \( p \geq 20 \). Therefore, this statement is false.

2. Behavior of \( m \): The function \( m(p) = 20(2)^p \) is an exponential function. As \( p \) increases, \( 2^p \) grows rapidly, meaning \( m(p) \) increases quickly at first because it's an exponential function. Thus, this statement is true.

3. Range: The minimum value of \( m \) occurs when \( p = 0 \): \[ m(0) = 20(2)^0 = 20. \] As \( p \) increases, \( m \) increases without bound. So, the range is \( m \geq 20 \). This statement is also true.

4. Decrease: As \( p \) increases, \( m \) does not decrease; it consistently increases. Hence, this statement is false.

Based on this analysis:

• The true statements are:
  • As p increases, m increases quickly at first and then slowly.
  • The range (m-values) includes the values (m≥20).

Therefore, the correct answer to select would be the statement about the behavior of \( m(p) \) as \( p \) increases: As p increases, m increases quickly at first and then slowly.