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Chapter 0: Problem 68

Quick Copy buys an office machine for 5200 dollars on January 1 of a given year. The machine is expected to last for 8 yr, at the end of which time its salvage value will be 1100 dollars. If the company figures the decline in value to be the same each year, then the book value, \(V(t),\) after \(t\) years, \(0 \leq t \leq 8,\) is given by $$V(t)=C-t\left(\frac{C-S}{N}\right)$$ where \(C\) is the original cost of the item, \(N\) is the number of years of expected life, and \(S\) is the salvage value. a) Find the linear function for the straight-line depreciation of the office machine. b) Find the book value after 0 yr, 1 yr, 2 yr, 3 yr, 4 yr, 7 yr, and 8 yr.

Short Answer

Expert verified
The linear function is \( V(t) = 5200 - 512.5t \). Book values: \( 5200, 4687.5, 4175, 3662.5, 3150, 1600.5, 1100 \).

Step by step solution

01

- Understand the formula

The formula for the book value after t years is given as \[ V(t) = C - t \left( \frac{C - S}{N} \right) \]where - \(C\) is the original cost (5200 dollars), - \(N\) is the number of years of expected life (8 years), and - \(S\) is the salvage value (1100 dollars).
02

- Substitute given values into the formula

Substitute the given values into the formula: \[ V(t) = 5200 - t \left( \frac{5200 - 1100}{8} \right) \]
03

- Simplify the depreciation factor

First, calculate the depreciation per year: \[ \frac{5200 - 1100}{8} = \frac{4100}{8} = 512.5 \]So, the formula becomes: \[ V(t) = 5200 - 512.5t \]
04

- Write the linear function

The linear function for the book value after t years is: \[ V(t) = 5200 - 512.5t \]
05

- Calculate book values for specific years

Use the linear function to find the book values for the specified years: \( t = 0, 1, 2, 3, 4, 7, 8 \)- For \( t = 0 \): \[ V(0) = 5200 - 512.5 \cdot 0 = 5200 \]- For \( t = 1 \): \[ V(1) = 5200 - 512.5 \cdot 1 = 4687.5 \]- For \( t = 2 \): \[ V(2) = 5200 - 512.5 \cdot 2 = 4175 \]- For \( t = 3 \): \[ V(3) = 5200 - 512.5 \cdot 3 = 3662.5 \]- For \( t = 4 \): \[ V(4) = 5200 - 512.5 \cdot 4 = 3150 \]- For \( t = 7 \): \[ V(7) = 5200 - 512.5 \cdot 7 = 1600.5 \]- For \( t = 8 \): \[ V(8) = 5200 - 512.5 \cdot 8 = 1100 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Straight-Line Depreciation
Straight-line depreciation is a method of allocating the cost of an asset evenly across its useful life. This means that the asset will lose the same amount of value each year until it reaches its salvage value. The main advantage of straight-line depreciation is its simplicity, making it easy to calculate and apply. Companies often use this method for financial reporting and tax purposes because it provides a clear and consistent expense over time.
Book Value
Book value represents the value of an asset as recorded in the company's books, reflecting its original cost minus accumulated depreciation. This value provides an estimate of what the asset is worth at a given point in time. In the given exercise, the book value formula after t years is


V(t) = 5200 - 512.5t
where
  • C is the original cost (5200 dollars)
  • P N is the number of years of expected life (8 years)
  • S is the salvage value (1100 dollars)

The book value decreases by the depreciation amount each year, reflecting the wear and tear or obsolescence of the asset.
Depreciation Formula
The depreciation formula in this context calculates the annual depreciation expense, which is the difference between the original cost and salvage value divided by the number of years of expected life.
Specifically, $$ \frac{C - S}{N} $$
Using the values from the exercise:

$$( \frac{5200 - 1100}{8} = 512.5 )$$
This means the machine depreciates by 512.5 dollars every year. To find the book value after t years, subtract the total depreciation (depreciation per year multiplied by the number of years) from the original cost. The formula to find the book value,

$$V(t) = C - t \times (\frac{C - S}{N})$$ $$V(t) = 5200 - 512.5t$$. helps in determining how much the machine is worth after any number of years from its purchase.

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