Problem 116 - Precalculus with Limits

Question

A tool and die company buys a machine for \(\$ 175,000\) and it depreciates at a rate of 30\(\%\) per year. In other words, at the end of each year the depreciated value is 70\(\%\) of what it was at the beginning of the year.) Find the depreciated value of the machine after 5 full years.

Step-by-Step Solution

Answer

The depreciated value of the machine after 5 years is calculated to be approximately \$36,302.50.

Step 1 - Determine The Depreciation Formula

The formula used to calculate depreciation in this scenario is \( D = P \times (1 - R)^N \), where \( D \) is the depreciated value, \( P \) is the principle amount (initial cost), \( R \) is the annual depreciation rate, and \( N \) is the number of years. The reason for this formula is that each year, the machine retains only a certain percentage (100% - depreciation rate) of its value.

Step 2 - Substitute The Given Values Into The Formula

We are given that \( P = \$175,000 \), \( R = 30\% = 0.3 \), and \( N = 5 \) years. Substituting these values into the formula gives: \( D = \$175,000 \times (1 - 0.3)^5\).

Step 3 - Calculate The Depreciated Value

Carry out the calculation in the formula to calculate \( D \), the depreciated value of the machine after 5 years.