An equation for the depreciation of a car is given by $y=A(1-r)^t$, where $y=$ current value, $A=$ original value, $r=$ rate of depreciation, and $t=$ time, in years. The current value of a car is $12,282.50. The car originally cost $20,000 and depreciates at a rate of 15% per year. How old is the car?
A. $\frac{1}{4}$ years
B. $\frac{1}{3}$ years
C. 3 years
D. 4 years
Answer (1)
Substitute the given values into the depreciation equation: 12282.50 = 20000 ( 1 − 0.15 ) t .
Simplify the equation: 0.614125 = ( 0.85 ) t .
Take the natural logarithm of both sides and use the power rule of logarithms: ln ( 0.614125 ) = t ln ( 0.85 ) .
Solve for t : t = l n ( 0.85 ) l n ( 0.614125 ) ≈ 3 . The car is 3 years old. 3
Explanation
Understanding the Problem We are given the depreciation equation y = A ( 1 − r ) t , where:
y is the current value of the car,
A is the original cost of the car,
r is the rate of depreciation per year,
t is the time in years. We are given the following values:
y = $12 , 282.50
A = $20 , 000
r = 15% = 0.15 We need to find the age of the car, which is t .
Substituting the Values Substitute the given values into the depreciation equation: 12282.50 = 20000 ( 1 − 0.15 ) t
Simplifying the Equation Simplify the equation: 12282.50 = 20000 ( 0.85 ) t
Isolating the Exponential Term Divide both sides by 20000: 20000 12282.50 = ( 0.85 ) t
Simplifying the Fraction Simplify the fraction: 0.614125 = ( 0.85 ) t
Taking the Natural Logarithm Take the natural logarithm of both sides: ln ( 0.614125 ) = ln (( 0.85 ) t )
Applying the Power Rule of Logarithms Use the power rule of logarithms: ln ( a b ) = b ln ( a ) ln ( 0.614125 ) = t ln ( 0.85 )
Solving for t Solve for t :
t = ln ( 0.85 ) ln ( 0.614125 )
Calculating the Value of t Calculate the value of t :
t = ln ( 0.85 ) ln ( 0.614125 ) ≈ 3
Final Answer Therefore, the age of the car is approximately 3 years.
Examples
Depreciation calculations are commonly used in business and finance to determine the reduction in value of an asset over time. For example, a company might use depreciation to estimate the value of its equipment or vehicles for accounting and tax purposes. Understanding depreciation can help businesses make informed decisions about when to replace assets and how to manage their finances effectively. Also, when you want to sell your used car, you can estimate its current value using the depreciation equation, which helps you set a reasonable selling price.
