Number of years 1 2 3 car 1 (value in dollars) 45,000 40,500 36,450 car 2 (value in dollars) 45,000 42,000 39,000 part a: what type of function, linear or exponential, can be used to describe the value of each of the cars after a fixed number of years? explain your answer. (2 points) part b: write one function for each car to describe the value of the car f(x), in dollars, after x years. (4 points) part c: summer wants to purchase a car that would have the greatest value in 14 years. will there be any significant difference in the value of either car after 14 years? explain your answer, and show the value of each car after 14 years. (4 points)

Part A:
The decline in the value of the first car is represented by an exponential function, since it does not have a constant rate of decrease as in the case of a linear function. The second car's depreciation in value may be modeled by a linear equation due to the constant difference between the yearly value.

Part B:
Let the function for car 1 be:
New value = original value * (depreciation factor)^x. For x = 1:
Depreciation factor = new value / original value
Depreciation factor = 40,500 / 45,000
Depreciation factor = 0.9

For car 1:
f(x) = 45,000(0.9)^x

For car 2:
Yearly decrease in value is $3,000, so:
f(x) = 45,000 - 3,000x

Part C:
We plug in 14 for x in both of our equations to find the value after 14 years of each car.

Car 1:
f(14) = 45,000(0.9)^14
f(14) = $10,294.56

Car 2:
f(14) = 45,000 - 3,000(14)
f(14) = $3,000

There is a significant difference between the values of the two cars after 14 years, greater than $7,000, so car 1 should be bought.