The value of a computer depreciates with age, as shown in the table.
Value of a Computer
| Age of computer (years) | Value of computer ($) |
| :---------------------- | :-------------------- |
| 0 | 1,600.00 |
| 1 | 1,280.00 |
| 2 | 1,024.00 |
| 3 | 819.20 |
| 4 | 655.36 |
Which function represents the value of a computer after [tex]$t$[/tex] years?
A. [tex]$f(t)=1,600-226.16 t$[/tex]
B. [tex]$f(t)=1,600-320 t$[/tex]
C. [tex]$f(t)=1,600(0.2)^t$[/tex]
D. [tex]$f(t)=1,600(0.8)^t$[/tex]
Answer (1)
Check if the function is linear by calculating the difference between consecutive values. Since the differences are not constant, the function is not linear.
Check if the function is exponential by calculating the ratio between consecutive values. The ratio is constant and equals 0.8.
The function is of the form f ( t ) = 1600 × ( 0.8 ) t .
The function that represents the value of the computer after t years is f ( t ) = 1 , 600 ( 0.8 ) t .
Explanation
Understanding the Problem We are given a table showing the value of a computer at different ages. We need to find a function that represents the value of the computer after t years.
Checking for Linearity First, let's check if the function is linear. If it is linear, the difference between consecutive values should be constant. The difference between the value at t = 0 and t = 1 is 1600 − 1280 = 320 . The difference between the value at t = 1 and t = 2 is 1280 − 1024 = 256 . Since the differences are not constant, the function is not linear.
Checking for Exponentiality Now, let's check if the function is exponential. If it is exponential, the ratio between consecutive values should be constant. The ratio between the value at t = 1 and t = 0 is 1600 1280 = 0.8 . The ratio between the value at t = 2 and t = 1 is 1280 1024 = 0.8 . The ratio between the value at t = 3 and t = 2 is 1024 819.20 = 0.8 . The ratio between the value at t = 4 and t = 3 is 819.20 655.36 = 0.8 . Since the ratios are constant, the function is exponential.
Finding the Exponential Function Since the function is exponential and the initial value (at t = 0 ) is 1600 , the function is of the form f ( t ) = 1600 × r t , where r is the common ratio. In this case, r = 0.8 , so the function is f ( t ) = 1600 ( 0.8 ) t .
Final Answer Comparing this with the given options, we see that the correct function is f ( t ) = 1 , 600 ( 0.8 ) t .
Conclusion The function that represents the value of the computer after t years is f ( t ) = 1 , 600 ( 0.8 ) t .
Examples
Understanding depreciation is crucial in finance and accounting. For instance, when a company buys equipment, its value decreases over time. Knowing the depreciation rate helps in calculating the asset's current value, which is essential for financial statements and tax purposes. This concept applies not only to computers but also to vehicles, machinery, and other assets that lose value as they age. By using an exponential decay model, businesses can accurately estimate the remaining value of their assets and make informed decisions about when to replace them.
