Mathematics: Two different cars each depreciate to 60% of their respective original values. the first car depreciates at an annual rate of10%. the second car depreciates at an annual rate of 15%. what is the approximate difference in the ages of the two cars? a. 1.7 yearsb. 2.0 yearsc. 3.1 yearsd. 5.0 years

5Step-by-step explanation:3x-7=8x=5...

Two different cars each depreciate to 60% of their

mxdv

12.03.2022

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.

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