Question

The value of a certain car after T years is modeled by the expression 15, 000(0.7)^t. What are the initial cost, I, and the rate of depreciation,r,of this car

Answer 1

Answer:

Initial cost (i) = 15,000

Rate of depreciation r = 0.3

Explanation:

To solve this we apply the exponential decay function

Y = i(1-r)^t

i = initial amount

r = depreciation/decay rate

t = time interval

y= 15000(0.7)^t

Where y = i(1-r)^t

Therefore from the equation

Initial cost (i) = 15,000

Rate of depreciation r =

1 - r = 0.7

r = 1 - 0.7

r = 0.3

Answer 2

a) V(0) = $15,000

b) r = -$5350.124*(0.7)^t

Explanation:

Given:

- The value of car as a function of time t in years:

V = 15,000*(0.7)^t

Find:

a) What are the initial cost V_I

b) the rate of depreciation r

Solution:

- The initial cost can be computed by setting t = 0 in the given relation. We will get the original cost of car before it started depreciating annually.

V(0) = 15,000*(0.7)^0

V(0) = $15,000

- The depreciation rate 'r' of the car can be evaluated by taking a derivative of V with respect to time t. That is rate of change of V with respect to time or the rate at which the value of car decreases:

dV/dt = r

r = d(15,000*(0.7)^t) / dt

r = 15,000*(0.7)^t * Ln (0.7)

r = -5350.124159*(0.7)^t

- We see that the depreciation rate r is also a function of time t in year. Every year the depreciation rate itself changes.