Question: The scrap value of a machine at the end of its useful life is given by \( S(n) = C(1 - r)^n \),…

The scrap value of a machine at the end of its useful life is given by \( S(n) = C(1 - r)^n \), where \( C \) is the original cost, \( n \) is the useful life of the machine in years, and \( r \) is the constant annual percentage of value lost. Find the scrap value of the following machine.

Original cost, $53,000; life, 8 years; annual rate of value lost, 10%

\( S = \$22,998.81 \)

Solution

The problem asks us to find the scrap value of a machine at the end of its useful life using the formula: \[ S(n) = C(1 - r)^n \] where \( C \) is the original cost, \( n \) is the useful life in years, and \( r \) is the annual rate of value lost. The given values are: - Original cost \( C = 53,000 \) - Useful life \( n = 8 \) years - Annual rate of value lost \( r = 0.10 \) (10%) Substitute the values into the formula: \[ S(8) = 53,000(1 - 0.10)^8 \] Calculate \( (1 - 0.10) \): \[ 1 - 0.10 = 0.90 \] Raise \( 0.90 \) to the power of \( 8 \): \[ 0.90^8 = 0.43046721 \] Multiply this result by the original cost: \[ S(8) = 53,000 \times 0.43046721 = 22,014.36 \] Therefore, the scrap value of the machine is approximately \$22,014.36.