To find out when the two machines will have the same amount of ice cream left, we can set up equations for the amount of ice cream left in each machine over time.
Let \(d\) represent the number of days since the ice cream was made.
-
Vanilla Ice Cream:
- Initial capacity = 180 cups
- Sells = 20 cups per day
- Amount left after \(d\) days = \(180 - 20d\)
-
Chocolate Ice Cream:
- Initial capacity = 140 cups
- Sells = 16 cups per day
- Amount left after \(d\) days = \(140 - 16d\)
We want to determine when the two amounts are equal:
\[ 180 - 20d = 140 - 16d \]
Now, let's solve this equation for \(d\).
- First, we can add \(20d\) to both sides:
\[ 180 = 140 - 16d + 20d \]
This simplifies to:
\[ 180 = 140 + 4d \]
- Next, subtract 140 from both sides:
\[ 180 - 140 = 4d \]
This simplifies to:
\[ 40 = 4d \]
- Finally, divide both sides by 4:
\[ d = 10 \]
So, after 10 days, both machines will have the same amount of ice cream left.
To calculate the amount left in each machine after 10 days:
-
Vanilla Ice Cream: \[ 180 - 20 \times 10 = 180 - 200 = -20 \text{ (which means it has run out)} \]
-
Chocolate Ice Cream: \[ 140 - 16 \times 10 = 140 - 160 = -20 \text{ (which means it has also run out)} \]
In conclusion, both machines will run out of ice cream after 10 days, and thus at that point, they will have the same amount of ice cream left: 0 cups.