To find when the two popcorn machines will have the same amount of popcorn left, we can set up equations for the amount of popcorn left in each theatre over time.
Let \( t \) represent the time in hours.
For Movie Theatre A:
- The initial capacity is 120 gallons.
- It sells 7 gallons of popcorn per hour.
The amount of popcorn left in Movie Theatre A after \( t \) hours can be expressed as: \[ P_A(t) = 120 - 7t \]
For Movie Theatre B:
- The initial capacity is 150 gallons.
- It sells 9 gallons of popcorn per hour.
The amount of popcorn left in Movie Theatre B after \( t \) hours can be expressed as: \[ P_B(t) = 150 - 9t \]
To find when the two amounts are equal, we set the two equations equal to each other: \[ 120 - 7t = 150 - 9t \]
Now, we can solve for \( t \):
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First, rearrange the equation: \[ 120 - 150 = -9t + 7t \] \[ -30 = -2t \]
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Divide by -2: \[ t = 15 \]
Thus, after 15 hours, both Movie Theatre A and Movie Theatre B will have the same amount of popcorn left.
Finally, let's verify the amount left in each theatre after 15 hours:
For Movie Theatre A: \[ P_A(15) = 120 - 7(15) = 120 - 105 = 15 \text{ gallons} \]
For Movie Theatre B: \[ P_B(15) = 150 - 9(15) = 150 - 135 = 15 \text{ gallons} \]
Both theatres will have 15 gallons of popcorn left after 15 hours, confirming that the solution is correct.