Question

The manufacturer's suggested retail price (MSRP) for a particular car is $25,125, and it is expected to be worth $19,645 in 2 years. (a) Find a linear depreciation function for this car. (b) Estimate the value of the car 3 years from now. (c) At what rate is the car depreciating? (a) What is the linear depreciation function for this car? f(x) = (Simplify your answer. Do not include the $ symbol in your answer.)

Please Answer a-b.​

Answer 1

The linear depreciation function for the car is f(x) = -2740x + 25125, which represents the value of the car at any given time x. After 3 years, the estimated value of the car is $17,905.

Step-by-step explanation:

To find a linear depreciation function for the car, we can use the formula for a straight line, which is f(x) = mx + b, where f(x) is the value of the car at time x, m is the slope of the line, and b is the y-intercept.

In this case, x represents the number of years after the purchase, m is the rate of depreciation, and b is the initial value of the car. Since the MSRP is $25,125 and the car is worth $19,645 after 2 years, we can calculate the annual depreciation rate (m) and determine the y-intercept (b).

The rate m is calculated as the change in value divided by the change in time, so m = ($19,645 - $25,125)/(2 - 0), which simplifies to m = -$5,480/2, giving us m = -$2,740 per year.

The y-intercept (b) is the initial value, which we know to be the MSRP of $25,125. Thus, the linear depreciation function is f(x) = -$2,740x + $25,125.

To estimate the value of the car 3 years from now, we use the function f(x) with x = 3. Therefore, f(3) = -$2,740(3) + $25,125, which equals $17,905.

Answer 2

a. Depreciation function

F = (1-r)^t

b, $17,357

c. 11.6%

Step-by-step explanation:

a. Firstly, we are told to find the linear depreciation function. This means an equation that shows how the depreciation is playing out.

To get this, we shall go the exponential way;

Let F = future value of the car, P = Present value of the car, r = percentage of depreciation and t = time or number of years

So the depreciation function will be;

F = P(1 - r)^t

c. Let’s calculate the rate at which it is depreciating

According to our formula, F = $19,645 while P = $25,125 and t = 3

The depreciation function is;

F = P(1-r)^t

So substituting these values, we have;

19,645 = 25,125(1-r)^2

Divide both sides by 25,125

19645/25125 = (1-r)^2

(1-r)^2 = 0.782

Find the square root of both sides;

1-r = √0.782

1-r = 0.884

r = 1-0.884 = 0.116 which is same as 11.6%

b. We want to find the cost of the car in 3 years

We use the same depreciation function;

F = P(1-r)^t

where in this case F = ? , P = 25,125 , r = 0.116 and t = 3

Substituting these values, we have

F = 25125(1-0.116)^3

F = 25125(0.884)^3 = $17,357