Question

The annual revenue of Amazon.com rose from approximately $10.7 billion in 2006 to $34.2 billion in 2010. Use this information to find both a linear model and an exponential model for Amazon.com's annual revenue I (in billions of dollars) as a function of time t in years since 2000.

Answer 1

1) Linear model

R(t) = y = at + b

Where t is the year - 2000 (year since 2000)

For year 2006, t = 6; for year 2010, t = 10

Then:

1) 10.7 = a*6 + b
2) 34.2 = a*10 + b

Subtract (1) from (2)

34.2 - 10.7 = 10a -6a

23.5 = 4a

a = 23.5/4 = 5.875

Now from (1) 10.7 = 6a + b => b = 10.7 - 6a = 10.7 - 6*5.875 = - 24.55

Then the resulting model is R(t) = 5.875a - 24.55 


2) Exponential model

R(t) = A[B]^t

(1) 10.7 = A[B]^6
(2) 34.2 = A[B]^10

Divide (2) by (1)

[34.2/10.7] = [B]^10 / [B]^6

3.1963 = [B]^4 => B = 1.3371

Now, from (1) 10.7 = A [1.3371]^6 => A = 10.7 / [1.337]^6 = 1.8725

Then, the model is R(t) = 1.8725{1.3371]^t