Question

Two different cars each depreciate to 60% of their respective original values. The first car depreciates at an annual rate of

Answer 1

Answer: A) 1.7 Years

Step-by-step explanation:

Answer 2

Answer:

A. 1.7 years

Step-by-step explanation:

Let $P_1$ be the original value of first car,

Since, the car depreciates at an annual rate of  10%,

Let after $t_1$ years the value of car is depreciated to 60%,

That is,

$P_1(1-\frac{10}{100})^{t_1}=60\%\text{ of }P_1$

$P_1(1-0.1)^{t_1}=0.6P_1$

$0.9^{t_1}=0.6$

Taking ln on both sides,

$t_1ln(0.9) = ln(0.6)$

$\implies t_1=\frac{ln(0.6)}{ln(0.9)}$

Now, let $P_2$ is the original value of second car,

Since, the car depreciates at an annual rate of 15%

Suppose after $t_2$ years it is depreciated to 60%,

$P_2(1-\frac{15}{100})^{t_2}=60\%\text{ of }P_2$

$P_2(1-0.15)^{t_2}=0.6P_2$

$0.85^{t_2}=0.6$

Taking ln on both sides,

$t_2ln(0.85) = ln(0.6)$

$\implies t_2=\frac{ln(0.6)}{ln(0.85)}$

$\because t_1-t_2=\frac{ln(0.6)}{ln(0.90)}-\frac{ln(0.6)}{ln(0.85)}$

$=1.70518303046$

$\approx 1.7$

Hence, the approximate difference in the ages of the two cars is 1.7 years,

Option 'A' is correct.