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

Initial cost (i) = 15,000

Rate of depreciation r = 0.3

Step-by-step 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

a) V (0) = $15,000

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

Step-by-step 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.