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

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

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 .

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 .

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

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

GPT-4o mini · Answer

To analyze the function $ m(p) = 20(2)^p $ where $ m $ represents the money and $ p $ represents the number of people who opt to pass it on, we can determine the following: 1. **Range of the function**: - The equation $ m(p) = 20(2)^p $ outputs values for money starting from $ m(0) = 20(2)^0 = 20 $. As $ p $ increases (meaning more people choose to pass it on), $ m $ grows exponentially without bound. Therefore, the range of $ m $ is $ m \geq 20 $. 2. **Domain of the function**: - The variable $ p $, representing the number of people passing on the money, can be any non-negative integer (0, 1, 2, ...). Thus, the domain is $ p \geq 0 $. 3. **Behavior of the function as $ p $ increases**: - As $ p $ increases, $ m $ increases quickly because every time $ p $ increases by 1, $ m $ is multiplied by 2. Thus, $ m $ grows rapidly, which means $ m $ increases quickly at first and then keeps increasing faster. Based on these analyses, we can evaluate the statements provided: - **The range (m-values) includes the values $ m \geq 20 $**. This is true. - **The domain (p-values) include the values greater than or equal to 20 (p ≥ 20)**. This is false; the domain is $ p \geq 0 $, not starting at 20. - **As p increases, m decreases slowly at first and then quickly**. This is false; $ m $ does not decrease; it increases. - **As p increases, m increases quickly at first and then slowly**. This is misleading because $ m $ does not increase slowly at first; it increases quickly and continues to increase quickly. In conclusion, the true statement is: - **The range (m-values) includes the values $ m \geq 20 $**.