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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 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.
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).
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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1 answer

The equation given is \( m(p) = 20(2^p) \). Let's analyze the statements one by one:

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

    • This is true for exponential functions of the form \( a(2^p) \). For low values of \( p \), \( m \) grows slowly, but as \( p \) increases, \( m \) accelerates rapidly. So, this statement is true.

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

    • Since \( m(p) = 20(2^p) \) and \( 2^p \) is always positive for all \( p \geq 0 \), therefore the minimum value of \( m \) when \( p = 0 \) is \( m(0) = 20(2^0) = 20 \). Thus, as \( p \) increases, \( m \) increases, meaning the range is indeed \( m \geq 20 \). This statement is also true.

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

    • This is not true because as \( p \) increases, \( m(p) \) consistently increases due to its exponential nature.

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

    • The domain of \( p \) is actually all non-negative integers (assuming \( p \) represents the number of people and can be 0 or more), so it is not correct to state the domain starts from 20. Therefore, this statement is false.

Thus, the true statements about the equation are:

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

If you need to select just one response, either the first or the second statement is acceptable, but the first option is often the most directly associated with the behavior of exponential growth.