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

Question

situation can be modeled by the exponential equation m (p) = 20(2)*
, 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)
• As p increases, m decreases slowly at first and then quickly.
• The range (m-values) includes the values (m ≥ 20).
O 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).

Answer 1

To evaluate the equation \( m(p) = 20(2^p) \) for Mason's study:

1. Understanding the function: The function describes how the amount of money \( m \) increases as more people \( p \) opt to pass the money on, doubling it each time.

2. Rate of increase: As \( p \) increases, \( m \) is given by \( 20(2^p) \). Since \( 2^p \) is an exponential function, it increases quickly as \( p \) gets larger. Thus, \( m \) does not decrease; instead, it increases.

3. Range of m: Since the smallest value of \( p \) is 0 (if the first person takes the money), the minimum value of \( m(0) = 20(2^0) = 20 \). As \( p \) increases, \( m \) increases without bound. Therefore, the range is \( m \geq 20 \).

4. Domain of p: The number of people \( p \) can take any non-negative integer value (0, 1, 2, ...), which means \( p \) is defined for all values \( p \geq 0 \).

Given these evaluations, we can analyze the provided statements:

• As p increases, m decreases slowly at first and then quickly. (False)
• The range (m-values) includes the values (m ≥ 20). (True)
• As p increases, m increases quickly at first and then slowly. (False)
• The domain (p-values) include the values greater than or equal to 20 (p ≥ 20). (False, because \( p \) starts at 0)

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