Use the image to answer the question.

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

Answer 1

To analyze the exponential equation \( m(p) = 20 \cdot (2^p) \), let’s break down the statements:

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

   • When \( p = 0 \), \( m(0) = 20 \). As \( p \) increases, \( 2^p \) grows rapidly, and therefore \( m \) will also increase significantly. Thus, for any value of \( p \) greater than 0, \( m \) will be greater than 20. Hence, the range includes all values \( m \geq 20 \) as \( p \) increases.

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

   • This statement is typical of exponential growth. Initially, as \( p \) increases from 0, \( m \) rises quickly because \( 2^p \) doubles for each increment in \( p \). However, the effective increase becomes less significant because the values of \( m \) are already getting significantly larger. This statement is also true.

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

   • This statement is false. The money \( m \) is not decreasing; it is an increasing function.

4. The domain (p-values) includes the values \( p \geq 20 \):

   • The domain of the function is not limited to values greater than or equal to 20; \( p \) can take on any non-negative value (including \( 0 \)). Therefore, this statement is false.

Considering the analysis above, the true statements are:

• The range (m-values) includes the values \( m \geq 20 \).
• As p increases, m increases quickly at first and then slowly.

Since the question asks for which statement is true, the first statement is the most straightforward correct answer regarding the range of the function.

The correct response is: The range (m-values) includes the values \( m \geq 20 \).