Question

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?

Answer 1

The car is 6.5 years old

Explanation:

An equation for the depreciation of a car is given by
`y = A(1 - r)^t`

y = current value of the car

A = original cost

r = rate of depreciation

t = time in years

The value of a car is half what it originally cost

So,
`y = (A)/(2)`

The rate of depreciation is 10% = 0.1 =r

Substitute the values in equation

`(A)/(2) = A(1 - 0.1)^t`

`(1)/(2) =(1 - 0.1)^t`

`(1)/(2) =(0.9)^t`

`0.5=0.9^t`

t=6.57

Hence The car is 6.5 years old

Answer 2

The car is about 6.6 years old.

Explanation:

Given : 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%.

To find : Approximately how old is the car?

Solution :

The value of a car is half what it originally cost i.e.
`y=(1)/(2) A`

The rate of depreciation is 10% i.e. r=10%=0.1

Substitute in the equation,
`y = A(1-r)^t`

`(1)/(2) A= A(1-0.1)^t`

`(1)/(2)= (0.9)^t`

Taking log both side,

`\log((1)/(2))=t\log (0.9)`

`t=(\log((1)/(2)))/(\log (0.9))`

`t=6.57`

`t\approx 6.6`

Therefore, the car is about 6.6 years old.