To find the value of the machine after 4 years given that the original cost is [tex]$2838 and the depreciation rate is 0.2 (or 20%), we can follow these steps:
1. Identify the given variables:
- \( C \) (original cost) = $[/tex]2838
- [tex]\( r \)[/tex] (rate of depreciation) = 0.2
- [tex]\( t \)[/tex] (time in years) = 4
2. Substitute these values into the depreciation formula:
[tex]\[
V = C(1 - r)^t
\][/tex]
Here,
[tex]\[
V = 2838 \cdot (1 - 0.2)^4
\][/tex]
3. Simplify the expression inside the parentheses:
[tex]\[
1 - 0.2 = 0.8
\][/tex]
So the formula becomes:
[tex]\[
V = 2838 \cdot (0.8)^4
\][/tex]
4. Calculate [tex]\((0.8)^4\)[/tex]:
[tex]\[
0.8^4 = 0.4096
\][/tex]
5. Multiply the original cost by this value:
[tex]\[
V = 2838 \cdot 0.4096
\][/tex]
6. Perform the multiplication:
[tex]\[
2838 \cdot 0.4096 = 1162.4448
\][/tex]
7. Round the result to the nearest cent:
[tex]\[
1162.4448 \approx 1162.44
\][/tex]
So, the value of the machine at the end of 4 years is [tex]\( \boxed{1162.44} \)[/tex] dollars.
