The general equation for depreciation is given by [tex]y=A(1-r)^t[/tex], where [tex]y=[/tex] current value, [tex]A=[/tex] original cost, [tex]r=[/tex] rate of depreciation, and [tex]t=[/tex] time, in years. A car was purchased 6 years ago for $25,000. If the annual depreciation rate is 11%, which equation can be used to determine the approximate current value of the car?
[tex]y=25,000(0.89)^6[/tex]
[tex]y=(25,000 \cdot 0.11)^6[/tex]
[tex]y=(25,000 \cdot 0.89)^6[/tex]
Answer (2)
Substitute the given values A = 25 , 000 , r = 0.11 , and t = 6 into the depreciation equation y = A ( 1 − r ) t .
Simplify the expression inside the parentheses: 1 − 0.11 = 0.89 .
The equation becomes y = 25 , 000 ( 0.89 ) 6 .
The equation that can be used to determine the approximate current value of the car is y = 25 , 000 ( 0.89 ) 6 .
Explanation
Understanding the Problem We are given the general equation for depreciation: y = A ( 1 − r ) t , where:
y is the current value,
A is the original cost,
r is the rate of depreciation,
t is the time in years.
We are given the following information:
The car was purchased 6 years ago, so t = 6 .
The original cost of the car was $25 , 000 , so A = 25 , 000 .
The annual depreciation rate is 11% , so r = 0.11 .
Objective We need to substitute the given values into the depreciation equation to find the equation that determines the approximate current value of the car.
Substitution Substitute the values A = 25 , 000 , r = 0.11 , and t = 6 into the equation y = A ( 1 − r ) t :
y = 25 , 000 ( 1 − 0.11 ) 6
Simplification Simplify the expression inside the parentheses: 1 − 0.11 = 0.89
Final Equation So the equation becomes: y = 25 , 000 ( 0.89 ) 6
This matches the first option provided.
Examples
Understanding depreciation is crucial in many real-life scenarios. For example, when buying a new car, its value decreases over time due to wear and tear, usage, and market conditions. The depreciation equation helps estimate the car's current value, which is essential for insurance purposes, selling the car, or understanding its financial impact. Similarly, businesses use depreciation to account for the declining value of assets like machinery and equipment, affecting their financial statements and tax obligations. By understanding depreciation, individuals and businesses can make informed decisions about investments, asset management, and financial planning.
To determine the current value of the car using the depreciation formula, we substitute the values into the equation, resulting in y = 25000 ( 0.89 ) 6 . This equation indicates how the car's value decreases over time due to depreciation. Hence, the correct option is y = 25000 ( 0.89 ) 6 .
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