Question

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)

Answer 1

what is the equation

Answer 2

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.

Answer 3

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.