Let's denote the amount of popcorn left in Movie Theatre A as \( A(t) \) and the amount of popcorn left in Movie Theatre B as \( B(t) \), where \( t \) is the number of hours since the machines were full.
For Movie Theatre A:
- Initial capacity: 120 gallons
- Rate of selling: 7 gallons per hour
The amount of popcorn left in Theatre A after \( t \) hours can be expressed as: \[ A(t) = 120 - 7t \]
For Movie Theatre B:
- Initial capacity: 150 gallons
- Rate of selling: 9 gallons per hour
The amount of popcorn left in Theatre B after \( t \) hours can be expressed as: \[ B(t) = 150 - 9t \]
Setting the two equations equal: To find out when the two theatres have the same amount of popcorn left, we need to set \( A(t) \) equal to \( B(t) \): \[ 120 - 7t = 150 - 9t \]
Solving for \( t \):
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Rearranging the equation: \[ 9t - 7t = 150 - 120 \] \[ 2t = 30 \]
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Dividing both sides by 2: \[ t = 15 \]
So, the two popcorn machines will have the same amount of popcorn left after 15 hours.
Verification:
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Amount of popcorn left in Theatre A after 15 hours: \[ A(15) = 120 - 7 \times 15 = 120 - 105 = 15 \text{ gallons} \]
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Amount of popcorn left in Theatre B after 15 hours: \[ B(15) = 150 - 9 \times 15 = 150 - 135 = 15 \text{ gallons} \]
Both theatres will have 15 gallons of popcorn left after 15 hours, confirming our solution.
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