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 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 . 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 .

Question

GPT-4o mini · Answer

Let's analyze the given equation $ m(p) = 20(2)^p $. 1. **Understanding how $ m $ changes with $ p $**: - The equation $ m(p) = 20(2)^p $ is an exponential function where the base is 2. - As $ p $ (the number of people who pass on the money) increases, $ m $ increases exponentially because $ 2^p $ grows very rapidly. - Therefore, as $ p $ increases, $ m $ increases quickly. It is accurate to say: - **As $ p $ increases, $ m $ increases quickly at first and then continues to increase rapidly.** 2. **Domain**: - The variable $ p $ represents the number of people who choose to pass the money on. Since it cannot be negative, the smallest value $ p $ can take is 0 (when the first person is asked). - Thus, the domain is $ p \geq 0 $. 3. **Range**: - The minimum value of $ m $ occurs when $ p = 0$: $$ m(0) = 20(2)^0 = 20. $$ - As $ p $ increases, $ m $ will also increase without bound, so the range of $ m $ is $ m \geq 20 $. Based on this analysis, we can conclude: - The statement that is true about the equation is: **As $ p $ increases, $ m $ increases quickly at first and then slowly.** For the other statements about the domain and range: - The domain (p-values) is $ p \geq 0 $. - The range (m-values) is $ m \geq 20 $. Thus, the complete true statements are: - **As $ p $ increases, $ m $ increases quickly at first and then slowly.** - **The range (m-values) includes the values $ m \geq 20 $.**