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The value of a machine, [tex]\( V \)[/tex], at the end of [tex]\( t \)[/tex] years is given by [tex]\( V = C(1 - r)^t \)[/tex], where [tex]\( C \)[/tex] is the original cost and [tex]\( r \)[/tex] is the rate of depreciation.

Find the value of a machine at the end of 4 years if the original cost was [tex]\( \$1972 \)[/tex] and [tex]\( r = 0.1 \)[/tex]. Round to the nearest cent.

Sagot :

GinnyAnsweridn
To find the value of the machine at the end of 4 years, we start with the formula for depreciation:

[tex]\[ V = C(1 - r)^t \][/tex]

where:
- [tex]\(V\)[/tex] is the value of the machine at the end of [tex]\(t\)[/tex] years,
- [tex]\(C\)[/tex] is the original cost of the machine,
- [tex]\(r\)[/tex] is the rate of depreciation,
- [tex]\(t\)[/tex] is the number of years.

Given:
- The original cost of the machine [tex]\(C\)[/tex] is \[tex]$1972, - The rate of depreciation \(r\) is 0.1 (or 10%), - The time period \(t\) is 4 years. Substituting these values into the formula, we get: \[ V = 1972 (1 - 0.1)^4 \] First, calculate the term inside the parentheses: \[ 1 - 0.1 = 0.9 \] Now raise this result to the power of 4: \[ 0.9^4 = 0.6561 \] Next, multiply this result by the original cost \(C\): \[ V = 1972 \times 0.6561 \] Performing the multiplication gives: \[ V = 1293.8292000000001 \] The value of the machine at the end of 4 years is approximately \$[/tex]1293.83 when rounded to the nearest cent.

Thus, the value of the machine at the end of 4 years is [tex]\(\boxed{1293.83}\)[/tex].
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