Question

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?
As p increases, m decreases slowly at first and then quickly.
As p increases, m increases quickly at first and then slowly.
The range (m-values) includes the values (m≥20)(m≥20).
The domain (p-values) include the values greater than or equal to 20 (p≥20)(p≥20).

Answer 1

To analyze the given situation described by the equation \( m(p) = 20(2)^p \):

1. **Understanding the function**:
- The function starts with \( m(0) = 20(2^0) = 20 \).
- Each time a person opts to double the money, the value of \( m \) doubles with respect to the number of people \( p \). Thus, as \( p \) increases, \( m \) increases rapidly because \( 2^p \) grows exponentially.

2. **Assessing the statements**:
- **First statement**: "As p increases, m decreases slowly at first and then quickly."
- This statement is **false**. As we established, \( m \) increases as \( p \) increases, not decreases.

- **Second statement**: "As p increases, m increases quickly at first and then slowly."
- This statement is **partially true** but not fully accurate regarding the behavior of exponential growth. Exponential growth actually increases quickly and continues to increase quickly as \( p \) increases.

- **Third statement**: "The range (m-values) includes the values (m ≥ 20)."
- This statement is **true**. The minimum value \( m \) can take is 20 (when \( p = 0 \)), and as \( p \) increases, \( m \) will only increase.

- **Fourth statement**: "The domain (p-values) includes the values greater than or equal to 20 (p ≥ 20)."
- This statement is **false**. The domain of \( p \) is typically non-negative integers (0, 1, 2, ...), meaning \( p \) can be 0 or any positive integer, not restricted to values greater than or equal to 20.

Based on this analysis, the correct statement is:

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