Question

6. A company car purchased for $39,600 depreciates at 12% per annum. What is the car

worth after 3 years?

Answer 1

$26,986.29

Explanation:

As the car's value depreciates at a constant rate of 12% per annum, we can use the exponential decay formula to create a function for the value of the car f(t) after t years.

Exponential Decay formula

`\boxed{f(t)=a(1-r)^t}`

where:

• f(t) is the value of the car (in dollars) after t years.
• a is the initial value of the car.
• r is the depreciation rate (as a decimal).
• t is the time period (number of years after purchase).

In this case, the initial value is $39,600, and the rate of depreciation is 12% per year. Therefore, the function that models the value of the car after t years is:

`f(t)=39600(1-0.12)^t`

`f(t)=39600(0.88)^t`

To calculate the value of the car after 3 years, substitute t = 3 into the function:

`\begin{aligned} f(3)&=39600(0.88)^3\\&=39600(0.681472)\\&=26986.2912\\&=26986.29\;(\sf 2\;d.p.)\end{aligned}`

Therefore, the car is worth $26,986.29 after 3 years.

Answer 2

$26,986.29

Explanation:

We can use the formula for calculating the depreciation of an asset over time:

wor

`\bold{D = P(1 - (r)/(100) )^t}`

where:

D= the current value of the asset

P = the initial purchase price of the asset

r = the annual depreciation rate as a decimal

t = the number of years the asset has been in use

In this case, we have:

P = $39,600

r = 12% = 0.12

t = 3 years

Substituting these values into the formula, we get:

`D= 39,600(1 - (12)/(100))^3\\D= 39,600(1 - 0.12)^3\\D= 39,600*0.88^3\\D= 39,600*0.681472\\D=26986.2912`

Therefore, the car is worth approximately $26,986.29 after 3 years of depreciation at a rate of 12% per annum.