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

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

GPT-4o mini · Answer

Let's analyze the equation $ m(p) = 20(2^p) $. 1. **Understanding the equation**: Here, $ m $ (the amount of money) starts at 20 and doubles each time $ p $ (the number of people who pass the money to the next individual) increases. The factor $ 2^p $ indicates that as more people choose to pass the money, the amount increases exponentially. 2. **Range of $ m $**: Since $ m = 20(2^p) $, and $ 2^p $ will always be greater than 1 for $ p \geq 0$, the minimum value of $ m $ occurs when $ p = 0$, which gives $ m = 20(2^0) = 20(1) = 20 $. As $ p $ increases, $ m $ becomes significantly larger due to the exponential growth of $ 2^p $. Therefore, the range of $ m $ includes values greater than 20: $ (m > 20) $ is true for $ p \geq 1 $. 3. **Behavior of $ m $**: As $ p $ increases, $ m $ increases exponentially due to the $ 2^p $ term. This means $ m $ will increase quickly at first (for small $ p $) and then continue to grow very quickly as $ p $ becomes larger, which contradicts the idea that it increases slowly at first then quickly. 4. **The domain of $ p $**: The variable $ p $, which represents the number of people who pass on taking the money, starts at 0 (the initial choice to take the money or pass it). Therefore, the domain should include non-negative integers, $ p \geq 0 $, not starting from $ p \geq 20 $. Based on the analysis above: - The statement "The range (m-values) includes the values (m > 20)" is true. - The statement "As p increases, m increases quickly at first and then slowly" is false. - The statement "As p increases, m decreases slowly at first and then quickly" is false. - The statement "The domain (p-values) include the values greater than or equal to 20 (p ≥ 20)" is false. **Conclusion**: The correct and true statement about the equation is that **the range (m-values) includes the values (m > 20)**.