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

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

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

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 increases quickly at first and then slowly.

GPT-4o mini · Answer

To analyze the equation $ m(p) = 20(2^p) $, we can look at the behavior of the function as $ p $ changes. 1. **As $ p $ increases, $ m $ behavior**: - The function $ 20(2^p) $ grows exponentially as $ p $ increases. - Therefore, as $ p $ increases, $ m $ does not decrease; instead, it increases quickly at first and continues to increase rapidly due to the exponential nature of $ 2^p $. 2. **Range of $ m $**: - The smallest value of $ m $ occurs when $ p = 0 $, giving $ m(0) = 20(2^0) = 20 $. As $ p $ increases, $ m $ increases infinitely. - Thus, the range of $ m $ is $ m \geq 20 $. 3. **Domain of $ p $**: - $ p $ represents the number of people who pass the money on, which can take values of 0 and greater (i.e., $ p \geq 0 $). - Therefore, the domain is $ p \geq 0 $. Based on this analysis, the true statement about the equation is: - **The range (m-values) includes the values (m≥20)**. The other options regarding the behavior of $ m $ as $ p $ increases and the values of $ p $ do not accurately describe the function.