An equation for the depreciation of a car is given by $y=A(1-r)^t$, where $y=$ current value of the car, $A=$ original cost, $r=$ rate of depreciation, and $t=$ time, in years. The value of a car is half what it originally cost. The rate of depreciation is 10\%. Approximately how old is the car?
A. 3.3 years
B. 5.0 years
C. 5.6 years
D. 6.6 years
Answer (2)
Substitute the given values into the depreciation equation: 2 A = A ( 1 − 0.10 ) t .
Simplify the equation: 2 1 = ( 0.9 ) t .
Take the natural logarithm of both sides: ln ( 2 1 ) = t ln ( 0.9 ) .
Solve for t : t = l n ( 0.9 ) l n ( 0.5 ) ≈ 6.6 .
The car is approximately 6.6 years old.
Explanation
Understanding the Problem We are given the depreciation equation y = A ( 1 − r ) t , where: y = current value of the car A = original cost r = rate of depreciation t = time in years
We are also given that the current value of the car is half its original cost, which means y = 2 A . The rate of depreciation is 10%, so r = 0.10 . We want to find the age of the car, which is t .
Substituting the Values Substitute the given values into the depreciation equation: 2 A = A ( 1 − 0.10 ) t
Simplifying the Equation Divide both sides of the equation by A :
2 1 = ( 1 − 0.10 ) t 2 1 = ( 0.9 ) t
Applying Logarithms Take the natural logarithm of both sides of the equation: ln ( 2 1 ) = ln ( ( 0.9 ) t ) Using the power rule of logarithms, we have: ln ( 2 1 ) = t ln ( 0.9 )
Isolating t Solve for t by dividing both sides by ln ( 0.9 ) :
t = ln ( 0.9 ) ln ( 2 1 )
Calculating t Calculate the value of t :
t = ln ( 0.9 ) ln ( 0.5 ) ≈ 6.5788
Final Answer The age of the car is approximately 6.58 years. Looking at the multiple-choice options, the closest value is 6.6 years.
Examples
Understanding depreciation is crucial in personal finance. For instance, when buying a new car, its value decreases over time due to wear and tear, technological advancements, and market conditions. The depreciation equation helps estimate the car's future value, aiding in decisions about when to sell or trade it in. This concept also applies to other assets like electronics and machinery, helping individuals and businesses make informed financial decisions.
The car is approximately 6.6 years old based on the depreciation formula. After calculating the time it takes for the car's value to decrease to half due to a 10% annual depreciation rate, we find that t is roughly 6.58 years. Hence, the closest option provided is 6.6 years.
;
