Movie Theatre A has a popcorn machine with a 100-gallon capacity, and sells 3 gallons of popcorn per hour. Movie Theatre B has a popcorn machine with a 150-gallon capacity, and sells 5 gallons of popcorn per hour. Write an equation to show when the two popcorn machines will have the same amount of popcorn left
Let's assume that the initial amount of popcorn in both machines is the same, and let's call that amount "P".
After "t" hours, the amount of popcorn left in Theatre A's machine can be calculated as:
P - 3t
Similarly, the amount of popcorn left in Theatre B's machine after "t" hours can be calculated as:
P - 5t
To find out when the two machines will have the same amount of popcorn left, we need to set these two expressions equal to each other:
P - 3t = P - 5t
Simplifying this equation, we get:
2t = P
Therefore, the two machines will have the same amount of popcorn left after "t" hours, where "t" is half of the initial amount of popcorn (P) divided by 2.
In other words, the equation to show when the two popcorn machines will have the same amount of popcorn left is:
t = P/4
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Let's assume "x" represents the number of hours elapsed since the popcorn machines were filled.
For Movie Theatre A, the amount of popcorn left after "x" hours can be represented as: 100 - 3x.
For Movie Theatre B, the amount of popcorn left after "x" hours can be represented as: 150 - 5x.
To find when both machines will have the same amount of popcorn left, we need to set the two equations equal to each other:
100 - 3x = 150 - 5x
Simplifying the equation:
3x - 5x = 150 - 100
-2x = 50
Dividing both sides by -2:
x = -50 / -2
x = 25
Therefore, both movie theatres will have the same amount of popcorn left after 25 hours.
To write an equation showing when the two popcorn machines will have the same amount of popcorn left, we can use the following variables:
- t: represents the number of hours since the popcorn machines started selling popcorn.
- A(t): represents the amount of popcorn remaining in Movie Theatre A's machine after t hours.
- B(t): represents the amount of popcorn remaining in Movie Theatre B's machine after t hours.
Given the information provided, the initial popcorn machine capacities are:
- Movie Theatre A: 100 gallons
- Movie Theatre B: 150 gallons
The amount of popcorn sold per hour by each machine is:
- Movie Theatre A: 3 gallons
- Movie Theatre B: 5 gallons
To determine the remaining amount of popcorn after t hours, we can use the equation:
A(t) = 100 - 3t
B(t) = 150 - 5t
In the equation for A(t), we subtract 3t from the initial capacity of 100 gallons to account for the 3 gallons sold per hour. Similarly, in the equation for B(t), we subtract 5t from the initial capacity of 150 gallons to account for the 5 gallons sold per hour.
Setting A(t) equal to B(t), we can solve for t to find when the two popcorn machines will have the same amount of popcorn left:
100 - 3t = 150 - 5t