Question

A car is sold for $16,000. after one year, the value of the car is $12,000. write an exponential function y to determine the value of the car after x years if the rate of decrease is the same each year. y = estimate the value of the car after 6 years. round the answer to the nearest dollar. $

Answer 1

The estimated value of a car that originally cost $16,000 and decreases in value by 25% annually is approximately $5,314 after 6 years.

Step-by-step explanation:

The question involves finding an exponential function to describe the depreciation of a car's value over time. Initially, the car is sold for $16,000 and after one year, its value decreases to $12,000. We can use this information to determine the annual rate of decrease and construct the exponential function.

Let the exponential function be y = a(1-r)^x, where a is the initial value, r is the rate of decrease, and x is the number of years. We know that a = $16,000 and after one year when x = 1, y = $12,000.

Finding the rate of decrease:

1. $12,000 = $16,000(1-r)^1
2. $12,000/$16,000 = (1-r)
3. (1-r) = 0.75
4. r = 0.25 (or 25% annual decrease)

We can now write the function as y = $16,000(1-0.25)^x.

To estimate the value of the car after 6 years, set x = 6 and calculate y:

1. y = $16,000(1-0.25)^6
2. y = $16,000(0.75)^6
3. y ≈ $5,314 (rounded to the nearest dollar)

Hence, the estimated value of the car after 6 years is approximately $5,314.