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 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 . 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.

Answer 1

To analyze the function \( m(p) = 20(2)^p \):

1. Behavior of m(p):

   • As \( p \) increases, \( (2)^p \) grows exponentially. This means that \( m(p) \) increases very quickly as \( p \) increases. So the statement about m increasing quickly at first and then slowly is incorrect.

2. Domain of p:

   • \( p \) represents the number of people who opt to pass the money on, so it must be a non-negative integer. Therefore, the correct domain is \( p \geq 0 \) (not \( p \geq 20 \)).

3. Range of m:

   • The minimum value of \( m \) occurs when \( p = 0 \): \( m(0) = 20(2)^0 = 20 \). As \( p \) increases, \( m(p) \) can take on any value greater than 20. Thus, the range is \( m \geq 20 \).

4. Behavior of m in relation to decreasing:

   • The statement about \( m \) decreasing is false because \( m(p) \) increases as \( p \) increases.

Bringing this all together:

• The correct statement regarding the equation is: The range (m-values) includes the values (m ≥ 20).