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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 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.

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

Answer 1

To analyze the equation \( m(p) = 20(2^p) \), we need to consider both the range and the behavior of the function as \( p \) (the number of people who pass on the money) increases.

1. Identifying the range: Since \( m(p) = 20(2^p) \), when \( p = 0 \), \( m(0) = 20(2^0) = 20 \). As \( p \) increases, \( 2^p \) grows exponentially, which means \( m(p) \) will also grow significantly. Therefore, the range of \( m \) can be expressed as \( m \geq 20 \) since it reaches its minimum value at \( 20 \) when no one passes it on (i.e., when \( p = 0 \)) and increases without bound as \( p \) increases.

2. Behavior as \( p \) increases:

   • As \( p \) increases, \( m(p) \) does not decrease; it increases exponentially. Specifically, \( m(p) \) increases quickly at first (because \( 2^p \) increases quickly), and it continues to increase even more rapidly as \( p \) increases further. Therefore, the statement about \( m \) decreasing is incorrect.

3. Domain: The domain of the function \( m(p) \) refers to the allowable values of \( p \). Since \( p \) represents the number of people who opt to pass on the money, it can take on non-negative integer values (0, 1, 2, 3, ...). Thus, the domain can be expressed as \( p \geq 0 \), not \( p \geq 20 \).

Based on this analysis, the statement that is true about the equation is:

The range (m-values) includes the values \( m \geq 20 \).