Let's denote the number of days that pass as \( x \).
For the vanilla ice cream machine:
- Initial capacity: 180 cups
- Daily sales: 20 cups
The amount of vanilla ice cream left after \( x \) days can be expressed as: \[ V(x) = 180 - 20x \]
For the chocolate ice cream machine:
- Initial capacity: 140 cups
- Daily sales: 16 cups
The amount of chocolate ice cream left after \( x \) days can be expressed as: \[ C(x) = 140 - 16x \]
We want to find the value of \( x \) when both machines have the same amount of ice cream left: \[ V(x) = C(x) \]
Now, we can set up the equation: \[ 180 - 20x = 140 - 16x \]
Next, we will solve for \( x \):
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Rearranging the equation gives: \[ 180 - 140 = 20x - 16x \] \[ 40 = 4x \]
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Dividing both sides by 4: \[ x = 10 \]
Now that we have \( x = 10 \), we can substitute this value back into either equation to find the amount of ice cream left in both machines after 10 days.
Calculating for the vanilla machine: \[ V(10) = 180 - 20(10) = 180 - 200 = -20 \]
Calculating for the chocolate machine: \[ C(10) = 140 - 16(10) = 140 - 160 = -20 \]
Thus, after 10 days, both ice cream machines will have -20 cups of ice cream left, indicating that both machines will be out of ice cream (and so the amount they have decreases to zero before this point).
The conclusion is that while they hit the same amount mathematically at 10 days, practically, they'll run out of ice cream before that day is fully realized. The key takeaway is they will both become empty at different times before reaching that 10-day point.
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