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

Question

GPT-4o mini · Answer

To analyze the equation $ m(p) = 20(2^p) $: 1. **Understanding the Function**: - The money $ m $ is calculated based on the number of people $ p $ who choose to pass the money on. - When $ p = 0 $, $ m(0) = 20(2^0) = 20 $. - When $ p = 1 $, $ m(1) = 20(2^1) = 40 $. - When $ p = 2 $, $ m(2) = 20(2^2) = 80 $, and so on. - As $ p $ increases, $ m $ doubles for each additional person who passes the money on, indicating an exponential growth. 2. **Behavior of the Function**: - As $ p $ increases, $ m $ increases quickly because it is an exponential function. - This means that for small values of $ p $, $ m $ increases relatively quickly (from 20 to 40 to 80), and as $ p $ continues to increase, the values of $ m $ get much larger (160, 320, etc.). 3. **Domain and Range**: - **Domain**: The number of people $ p $ can start from 0 and go up to any non-negative integer, so $ p \geq 0 $. - **Range**: Since $ m(p) $ starts at 20 and can increase indefinitely, the range of $ m $ is $ m \geq 20$. Considering the statements provided: - **Statement 1**: "As $ p $ increases, $ m $ increases quickly at first and then slowly." - This is incorrect; $ m $ continues to increase quickly as $ p $ increases. - **Statement 2**: "As $ p $ increases, $ m $ decreases slowly at first and then quickly." - This is incorrect; $ m $ does not decrease at all. - **Statement 3**: "The range (m-values) includes the values $ m \geq 20 $." - This is true since $ m $ starts at 20 and increases without bound. - **Statement 4**: "The domain (p-values) includes the values $ p \geq 20 $." - This is incorrect; $ p $ can start from 0. Therefore, the true statement is: **The range (m-values) includes the values $ m \geq 20 $.**