The equation \( y = 3(2)^t \) models the number of people who visited a haunted house since it opened, where:
- \( y \) represents the number of visitors,
- \( t \) represents the time in hours since the haunted house opened.
Here's an analysis of this model:
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Initial Value: At \( t = 0 \), \[ y = 3(2)^0 = 3 \times 1 = 3. \] This means that initially, 3 people visited the haunted house when it first opened.
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Growth Rate: The equation indicates that the number of visitors doubles every hour (since the base of the exponent is 2). This means after each hour, the number of visitors is multiplied by 2 compared to the previous hour.
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Visitor Count Over Time:
- At \( t = 1 \): \[ y = 3(2)^1 = 3 \times 2 = 6. \]
- At \( t = 2 \): \[ y = 3(2)^2 = 3 \times 4 = 12. \]
- At \( t = 3 \): \[ y = 3(2)^3 = 3 \times 8 = 24. \]
- At \( t = 4 \): \[ y = 3(2)^4 = 3 \times 16 = 48. \]
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Exponential Growth: This model reflects exponential growth, where the number of visitors increases rapidly as time progresses.
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Graphing the Function: If you were to graph this function, you would see a curve that starts at \( (0, 3) \) and rises steeply as \( t \) increases.
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Predictions: You can use this model to predict the number of visitors at any given time by substituting the desired hour into the equation.
If you have specific questions or if there's a particular aspect of this model you'd like to explore further, let me know!
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