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?

Question

(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).

Answer 1

Let's analyze the given equation \( m(p) = 20(2)^p \).

1. As \( p \) increases, \( m \) increases quickly at first and then slowly: This is false. The function \( m(p) = 20(2)^p \) is an exponential function, which means it increases rapidly as \( p \) increases. There is no "slow" increase; instead, it increases increasingly more rapid.

2. The range (m-values) includes the values \( m \geq 20 \): This is true. When \( p = 0 \), \( m(0) = 20(2)^0 = 20 \). As \( p \) increases, \( m \) will only get larger because \( 2^p \) grows exponentially. Thus, all values of \( m \) will be greater than or equal to 20.

3. As \( p \) increases, \( m \) decreases slowly at first and then quickly: This is false. The function does not decrease; it increases as \( p \) increases.

4. The domain (p-values) include the values greater than or equal to 20 (p ≥ 20): This is false. The domain of \( p \) is all real numbers (typically non-negative integers in practical scenarios for this type of problem), but does not start at 20. It can be \( p = 0, 1, 2, ...\) and so on.

Conclusion:

The correct statement among the options is: The range (m-values) includes the values \( m \geq 20 \).