To find the value [tex]\( y \)[/tex] of Ty's computer after [tex]\( x \)[/tex] years, where the computer depreciates at a rate of [tex]\( 12\% \)[/tex] per year, we need to use the depreciation formula. Depreciation at a constant percentage rate indicates exponential decay.
The value of the computer after [tex]\( x \)[/tex] years can be represented by the equation:
[tex]\[ y = P \times (1 - r)^x \][/tex]
where:
- [tex]\( y \)[/tex] is the value of the computer after [tex]\( x \)[/tex] years,
- [tex]\( P \)[/tex] is the initial value of the computer,
- [tex]\( r \)[/tex] is the depreciation rate,
- [tex]\( x \)[/tex] is the number of years.
Given:
- [tex]\( P = 499 \)[/tex] (the initial value of the computer in dollars),
- [tex]\( r = 0.12 \)[/tex] (the depreciation rate as a decimal).
Therefore, the value [tex]\( y \)[/tex] of Ty's computer after [tex]\( x \)[/tex] years is:
[tex]\[ y = 499 \times (1 - 0.12)^x \][/tex]
Simplifying the equation, we have:
[tex]\[ y = 499 \times 0.88^x \][/tex]
Thus, the equation for the value of his computer [tex]\( x \)[/tex] years after he bought it is:
[tex]\[ y = 499 \times 0.88^x \][/tex]
This equation shows how the computer's value decreases exponentially over time with an annual depreciation rate of [tex]\( 12\% \)[/tex].
