Question

Question

Use the image to answer the 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

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

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 .

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 .
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Answer 1

The exponential equation given is \( m(p) = 20(2^p) \), where \( m \) represents money and \( p \) is the number of people who opt to pass the money on.

Let's analyze the statements provided:

1. As \( p \) increases, \( m \) increases quickly at first and then slowly.
   This statement is false because as \( p \) increases, \( m(p) \) increases exponentially. It increases quickly and continues to increase quickly without slowing down in terms of the growth rate.

2. As \( p \) increases, \( m \) decreases slowly at first and then quickly.
   This statement is also false because \( m(p) \) does not decrease; it only increases as \( p \) increases.

3. The range (m-values) includes the values \( m \geq 20 \).
   This statement is true. The minimum value occurs when \( p = 0 \), resulting in \( m(0) = 20(2^0) = 20 \). Since the function \( m(p) \) only increases from there, the range does indeed include all values \( m \geq 20 \).

4. The domain (p-values) include the values \( p \geq 20 \).
   This statement is false because there is no restriction on \( p \); it can be any non-negative integer (0, 1, 2, ...).

Based on the analysis, the correct and true statement about the equation is:

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