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Chapter 9: Problem 72

Depreciation A company buys a machine for \(\$ 475,000\) that depreciates at a rate of 30\(\%\) per year. Find a formula for the value of the machine after \(n\) years. What is its value after 5 years?

Short Answer

Expert verified
The machine would be worth approximately \$104,440.5 after 5 years. The general formula for the value of the machine after \(n\) years is \(V_{n}= 475,000 * (0.7)^{n}\).

Step by step solution

01

Understand the Information Given

The machine was initially bought for \$475,000 and depreciates at a rate of 30\% yearly. The task is to find a formula to represent the value of the machine after \(n\) years and calculate its value after 5 years.
02

Find the Formula for Depreciation

The formula commonly used to calculate depreciation is: \(V_{n}= V_{0} * (1 - r)^{n}\) where: \(\n\) \(V_{n}\) is the value of the asset after \(n\) years, \(\n\) \(V_{0}\) is the initial value of the asset, \(\n\) \(r\) is the rate of depreciation, \(\n\) \(n\) is the number of years.
03

Substitute the Values into the Formula

Substituting the given values into the formula, the formula becomes: \(V_{5}= 475,000 * (1 - 0.3)^{5}\)
04

Calculate the Value

Evaluate the expression: \(\n\) \(V_{5}= 475,000 * (0.7)^{5}\) \(\n\) \(V_{5}= \$104,440.5\)
05

Formulate the General Formula

The general formula for the value of the machine after \(n\) years will be: \(\n\) \(V_{n}= 475,000 * (0.7)^{n}\)

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