Question

Suppose a car that sells for $40,000 depreciates 10% per year. How many

Answer 1

Answer:

It would take 5 years for the car to have a value of less than $25,000

Step-by-step explanation:

Exponential Decaying Model

The exponential function is often used to model natural decaying processes, where the change is proportional to the actual quantity.

We have the initial value of a car is $40,000. Each year it depreciates by 10%.

Thus the first year its value is 90% of the initial value:

V1 = 90 * $40,000 / 100 = $36,000

By the second year its value is 90% of $36,000:

V2 = 90 * $36,000 / 100 = $32,400

Note the value for a year n is the original value multiplied by 90% (or 0.9) to the power of n:

$Vn = 40,000 \cdot 0.9^n$

To find the number of years needed to have a value of less than $25,000, we solve the equation:

$40,000 \cdot 0.9^n = 25,000$

Dividing by 40,000:

$0.9^n = 25,000/40,000 = 0.625$

Taking logarithms:

$n\log 0.9=\log 0.625$

$n=\log 0.625 / \log 0.9$

n =4.5

We'll round up to n = 5

It would take 5 years for the car to have a value of less than $25,000