The given exponential equation \( m(p) = 20(2^p) \) models the amount of money \( m \) that Mason has after \( p \) people decide to pass on the money to the next person instead of taking it. Here, \( 20 \) is the initial amount of money, and \( 2^p \) indicates that the amount doubles with each person who passes it on.
From the equation, we can analyze a few statements:
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Initial amount: When \( p = 0 \) (the first person), \( m(0) = 20(2^0) = 20 \). This means the initial amount is indeed $20.
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Exponential growth: As more people pass on the money, the amount of money increases exponentially. For every additional person \( p \) who passes it on, the total amount of money doubles.
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Rapid increase: After a few iterations, the amount of money can grow significantly. For example:
- If \( p = 1 \): \( m(1) = 20(2^1) = 40 \)
- If \( p = 2 \): \( m(2) = 20(2^2) = 80 \)
- If \( p = 3 \): \( m(3) = 20(2^3) = 160 \)
- If \( p = 4 \): \( m(4) = 20(2^4) = 320 \)
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Limiting condition: The process continues until someone decides to take the money, meaning that \( p \) will be some finite integer when the money is finally accepted.
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Outcome: If \( p \) becomes very large, \( m(p) \) can represent a very large amount of money (as it approaches infinity in theory).
Therefore, the statements that could be true about the equation might include the facts about exponential growth, the doubling effect for each person who passes it on, and how the initial amount is $20, among others.
If you have specific statements to evaluate, please share them, and I can help determine which are true based on the equation.
3 answers