To find the number of days \( x \) after which the two machines will have the same amount of ice cream left, we can set up the following equations based on the initial capacities and daily sales.
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Vanilla Ice Cream Machine:
- Initial capacity: 180 cups
- Sells 20 cups per day
- Amount of ice cream left after \( x \) days: \[ 180 - 20x \]
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Chocolate Ice Cream Machine:
- Initial capacity: 140 cups
- Sells 16 cups per day
- Amount of ice cream left after \( x \) days: \[ 140 - 16x \]
To find when both machines have the same amount of ice cream left, we can set the two expressions equal to each other: \[ 180 - 20x = 140 - 16x \]
Now, we will solve for \( x \).
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Rearranging the equation: \[ 180 - 140 = 20x - 16x \] \[ 40 = 4x \]
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Dividing both sides by 4: \[ x = 10 \]
So, after 10 days, both machines will have the same amount of ice cream left.
To confirm, we can substitute \( x = 10 \) back into both equations to find the amount of ice cream left in each machine:
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For the vanilla machine: \[ 180 - 20(10) = 180 - 200 = -20 \text{ (which means it will be out of ice cream before this time)} \]
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For the chocolate machine: \[ 140 - 16(10) = 140 - 160 = -20 \text{ (meaning it will also be out of ice cream)} \]
Thus, the answer is validated by the calculation leading to \( x = 10 \) days being the point in time when both machines run out of ice cream, thereby matching their remaining amounts at zero eventually after continuous sales.