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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 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 .

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)
.

1 answer

The equation given is \(m(p) = 20(2)^p\). Let's analyze it:

  1. As \(p\) (the number of people who opt to pass) increases, the term \(2^p\) increases exponentially. Therefore:

    • As \(p\) increases, \(m\) increases quickly because you are doubling the amount each time a person passes the money on.

  2. Regarding the range (m-values):

    • The minimum value of \(m\) occurs when \(p = 0\), which gives \(m(0) = 20(2)^0 = 20(1) = 20\). As \(p\) increases, \(m\) will continue to grow without bound, making the range \(m \geq 20\).

  3. Regarding the domain (p-values):

    • The number of people passing on the money, \(p\), can start at 0 and can increase indefinitely (there is no upper limit). Therefore, the domain is \(p \geq 0\).

Based on this analysis, the correct statements are:

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

Ultimately, the true statement regarding the equation is:

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

The best response among your options would be:
The range (m-values) includes the values \(m \geq 20\).