Question
An increasing curve with an arrow on the upper end is plotted on a coordinate plane. The x-axis is labeled people and ranges from 0 to 5 in increments of 1. The y-axis is labeled money and ranges from 0 to 500 in increments of 100.
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 .
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 .
As p increases, m increases quickly at first and then slowly.
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 .
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 .
As p increases, m increases quickly at first and then slowly.
Answers
GPT-4o mini
To analyze the given exponential equation \( m(p) = 20(2^p) \), let's break down the statements.
1. **Understanding the function**:
- The formula \( m(p) = 20(2^p) \) shows that as \( p \) (the number of people passing on the money) increases, \( m \) (the amount of money) doubles for each additional person. This means \( m \) grows exponentially.
2. **Analyzing the statements**:
- **Statement 1**: "The range (m-values) includes the values (m ≥ 20)."
- This is correct. When \( p = 0 \), \( m = 20(2^0) = 20 \). As \( p \) increases, \( m \) increases without bound. So, the range is indeed \( m \geq 20 \).
- **Statement 2**: "As p increases, m decreases slowly at first and then quickly."
- This is incorrect. \( m \) increases as \( p \) increases, not decreases.
- **Statement 3**: "The domain (p-values) include the values greater than or equal to 20 (p ≥ 20)."
- This is incorrect. The domain is all non-negative integers (since \( p \) can be 0, 1, 2, ..., and so on), not starting from 20.
- **Statement 4**: "As p increases, m increases quickly at first and then slowly."
- This is incorrect for this equation. The growth is exponential, meaning that it increases rapidly as \( p \) increases.
So, the only true statement is:
**"The range (m-values) includes the values (m ≥ 20)."**
1. **Understanding the function**:
- The formula \( m(p) = 20(2^p) \) shows that as \( p \) (the number of people passing on the money) increases, \( m \) (the amount of money) doubles for each additional person. This means \( m \) grows exponentially.
2. **Analyzing the statements**:
- **Statement 1**: "The range (m-values) includes the values (m ≥ 20)."
- This is correct. When \( p = 0 \), \( m = 20(2^0) = 20 \). As \( p \) increases, \( m \) increases without bound. So, the range is indeed \( m \geq 20 \).
- **Statement 2**: "As p increases, m decreases slowly at first and then quickly."
- This is incorrect. \( m \) increases as \( p \) increases, not decreases.
- **Statement 3**: "The domain (p-values) include the values greater than or equal to 20 (p ≥ 20)."
- This is incorrect. The domain is all non-negative integers (since \( p \) can be 0, 1, 2, ..., and so on), not starting from 20.
- **Statement 4**: "As p increases, m increases quickly at first and then slowly."
- This is incorrect for this equation. The growth is exponential, meaning that it increases rapidly as \( p \) increases.
So, the only true statement is:
**"The range (m-values) includes the values (m ≥ 20)."**