Question
Joe is doing a study, he begins with $20 and as individuals on the street if they want the money or if they want him to double it and give it to the next person. He continues to do this until someone takes the money. It can be modeled by the 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 statement is true
As p increases, m increases quickly at first and then slowly
The domain (p-values) include the values greater than or equal to 20(p>=20)
As p increases m decrease slowly at first and then quickly
The range (m-values) includes (m>=20)
As p increases, m increases quickly at first and then slowly
The domain (p-values) include the values greater than or equal to 20(p>=20)
As p increases m decrease slowly at first and then quickly
The range (m-values) includes (m>=20)
Answers
GPT-4o mini
Let's analyze the function \( m(p) = 20 \cdot (2)^p \):
1. **Understanding the function**:
- The function is an exponential function, where \( p \) represents the number of people who passed the money along.
- \( m(p) \) represents the amount of money Joe has after \( p \) people passed the money.
2. **Behavior of the function as \( p \) increases**:
- As \( p \) increases, \( (2)^p \) grows exponentially. This means that \( m(p) = 20 \cdot (2)^p \) increases quickly.
- There is no point at which \( m \) would start increasing slowly, as exponential growth continuously accelerates.
3. **Domain**:
- The domain of \( p \) (the number of people) is \( p \geq 0 \) because there can’t be a negative number of people.
4. **Range**:
- The minimum value of \( m(0) \) is 20. As \( p \) increases, \( m \) increases without bound. Therefore, the range of \( m \) is \( m \geq 20 \).
Now, let's evaluate each statement:
1. **"As p increases, m increases quickly at first and then slowly"**:
- This statement is **false**. \( m(p) \) increases quickly and continues to increase quickly due to the exponential nature of the function.
2. **"The domain (p-values) include the values greater than or equal to 20 (p >= 20)"**:
- This statement is **false**. The domain is \( p \geq 0 \).
3. **"As p increases m decreases slowly at first and then quickly"**:
- This statement is **false**. \( m(p) \) does not decrease; it increases.
4. **"The range (m-values) includes (m >= 20)"**:
- This statement is **true**. The range of \( m(p) \) begins at 20 and increases without bound.
Thus, the true statement is:
**The range (m-values) includes (m >= 20)**.
1. **Understanding the function**:
- The function is an exponential function, where \( p \) represents the number of people who passed the money along.
- \( m(p) \) represents the amount of money Joe has after \( p \) people passed the money.
2. **Behavior of the function as \( p \) increases**:
- As \( p \) increases, \( (2)^p \) grows exponentially. This means that \( m(p) = 20 \cdot (2)^p \) increases quickly.
- There is no point at which \( m \) would start increasing slowly, as exponential growth continuously accelerates.
3. **Domain**:
- The domain of \( p \) (the number of people) is \( p \geq 0 \) because there can’t be a negative number of people.
4. **Range**:
- The minimum value of \( m(0) \) is 20. As \( p \) increases, \( m \) increases without bound. Therefore, the range of \( m \) is \( m \geq 20 \).
Now, let's evaluate each statement:
1. **"As p increases, m increases quickly at first and then slowly"**:
- This statement is **false**. \( m(p) \) increases quickly and continues to increase quickly due to the exponential nature of the function.
2. **"The domain (p-values) include the values greater than or equal to 20 (p >= 20)"**:
- This statement is **false**. The domain is \( p \geq 0 \).
3. **"As p increases m decreases slowly at first and then quickly"**:
- This statement is **false**. \( m(p) \) does not decrease; it increases.
4. **"The range (m-values) includes (m >= 20)"**:
- This statement is **true**. The range of \( m(p) \) begins at 20 and increases without bound.
Thus, the true statement is:
**The range (m-values) includes (m >= 20)**.