Question

The value of Sara's new car decreases at a rate of 8% each year. Write an exponential function to model the decrease in the car's value each month.

Answer 1

Answer:

Example: Value = 20,000 (2.98/3)ⁿ

Explanation:

The function of a decay exponential function with decreasing rate r is:

y = A (1 - r)ⁿ

For example: y = 10(1 - 0.1)ⁿ, is a exponential function with decreasing rate 0.1 or 10%.

Then, for the given decreasing rate of 8% yearly, we fiirst find the monthly rate by dividing by 12: r = 8% / 12 = 0.08 / 12 = 0.02 / 3.

With that, the general form of the searched function to model the decrease in the car's value eah month is:

Value = A (1 - 0.02/3)ⁿ = A (2.98 / 3)ⁿ

In that equation, A is the initial value of the car.

Suppose a car whose initial value is $ 20,000, then the function (model) is:

Value = 20,000 (2.98/3)ⁿ

You van verify the validity of that model by doing a table:

Month          value 20,000(2.98/3)ⁿ

0                   20,000 (2.98 / 3)⁰ = 20,000
1                   20,0000 (2.98 / 3) = 19,866.67
3                   20,000 (2.98 / 3)² = 19,734.22
4                   20,000 (2.98 / 3)³ = 19,602.66

And now calculate the rate of decrease of the value for any consecutive pair of months.

For example for months 3 and 4, rate of decrease = [19,734.22 - 19,602.66] / 19,734.22 =0.006667 monthly.

Multiply by 12 to find the rate per year: 0.006667 (12) = 0.08 = 8%