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].
