Movie Theatre A has a popcorn machine with a 120-gallon capacity. It sells 7 gallons of popcorn per hour. Movie Theatre B has a popcorn machine with a 150-gallon capacity. It sells 9 gallons of popcorn per hour. Write and solve an equation to show when the two popcorn machines will have the same amount of popcorn left.(1 point)

3 answers

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 \( t \):

  1. Rearranging the equation: \[ 9t - 7t = 150 - 120 \] \[ 2t = 30 \]

  2. Dividing both sides by 2: \[ t = 15 \]

So, the two popcorn machines will have the same amount of popcorn left after 15 hours.

Verification:

  • Amount of popcorn left in Theatre A after 15 hours: \[ A(15) = 120 - 7 \times 15 = 120 - 105 = 15 \text{ gallons} \]

  • 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.

what is the equation

The equation that represents when the two popcorn machines will have the same amount of popcorn left is:

\[ 120 - 7t = 150 - 9t \]

Where:

  • \( 120 - 7t \) is the amount of popcorn left in Movie Theatre A after \( t \) hours,
  • \( 150 - 9t \) is the amount of popcorn left in Movie Theatre B after \( t \) hours.

By solving this equation, we found that \( t = 15 \) hours, which is the time when both machines will have the same amount of popcorn left.

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