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A small business has just invested in a new piece of equipment. The business's accountant estimates that the value of the equipment will decrease over time at a constant rate. Based on this assumption, the accountant properly estimates the value of the equipment 3 years afterward to be \(\$\mathrm{X}\), 6 years afterward to be \(\$\mathrm{Y}\), and 8 years afterward to be, for the first time, \($0\).
Select values for \(\mathrm{X}\) and for \(\mathrm{Y}\) that are jointly consistent with the given information. Make only two selections, one in each column.
X
Y
200
300
400
500
600
This is a time-based depreciation problem, so we'll use a timeline showing the equipment value over 8 years.
Timeline Visualization:
Year: 0-------3-------6-------8
Value: V₀ X Y 0
|<--3d-->|<--3d-->|<--2d-->|
Where:
Since depreciation is constant:
From our timeline:
From the third equation: \(\mathrm{V_0 = 8d}\)
Substituting back:
Therefore: \(\mathrm{X/Y = 5d/2d = 2.5}\)
This means \(\mathrm{X = 2.5Y}\)
We need Y such that 2.5Y is also in our answer choices.
If \(\mathrm{Y = 200 \rightarrow X = 2.5 \times 200 = 500}\)
Is 500 in our choices? Yes! ✓
Stop here - we found our answer.
With X = 500 and Y = 200:
Final Answer:
These values satisfy the constant depreciation requirement where the equipment depreciates $100 per year from an initial value of $800.