Let's begin by finding the linear and exponential models for Amazon.com's annual revenue $I$ as a function of time $t$ in years since 2000. We are given:

• In 2006 ($t = 6$), the revenue was $10.7$ billion dollars.
• In 2010 ($t = 10$), the revenue was $34.2$ billion dollars.

Part (a) - Linear Model

The general form of a linear model is:

$I(t) = mt + b$

where $m$ is the slope, and $b$ is the intercept.

Step 1: Calculate the slope $m$

We can calculate the slope $m$ as:

$m = \frac{I(10) - I(6)}{10 - 6} = \frac{34.2 - 10.7}{10 - 6} = \frac{23.5}{4} = 5.875$

So, $m = 5.875$.

Step 2: Find the intercept $b$

To find $b$, we can use one of the points, say $(6, 10.7)$:

$10.7 = 5.875(6) + b$ $10.7 = 35.25 + b$ $b = 10.7 - 35.25 = -24.55$

Thus, the linear model is:

$I(t) = 5.88t - 24.6$

(Rounding to three significant digits)

Part (b) - Exponential Model

The general form of an exponential model is:

$I(t) = I_0 e^{kt}$

where $I_0$ is the initial value (at $t = 0$), and $k$ is the growth rate.

Step 1: Write two equations using the given data

We know:

• $I(6) = 10.7$
• $I(10) = 34.2$

This gives us two equations:

$10.7 = I_0 e^{6k}$ $34.2 = I_0 e^{10k}$

Step 2: Solve for $k$

Dividing the second equation by the first equation:

$\frac{34.2}{10.7} = \frac{I_0 e^{10k}}{I_0 e^{6k}} \implies \frac{34.2}{10.7} = e^{4k}$ $e^{4k} = \frac{34.2}{10.7} = 3.196$ $4k = \ln(3.196) \implies k = \frac{\ln(3.196)}{4} = \frac{1.162}{4} = 0.2905$

Thus, $k = 0.2905$.

Step 3: Solve for $I_0$

Substitute $k = 0.2905$ into one of the original equations, say $10.7 = I_0 e^{6k}$:

$10.7 = I_0 e^{6 \times 0.2905} = I_0 e^{1.743}$ $10.7 = I_0 \times 5.717$ $I_0 = \frac{10.7}{5.717} = 1.872$

Thus, the exponential model is:

$I(t) = 1.87 e^{0.291t}$

(Rounding to three significant digits)

Summary:

• Linear model: $I(t) = 5.88t - 24.6$
• Exponential model: $I(t) = 1.87 e^{0.291t}$

Would you like more details on any part of the solution?

5 Related Questions:

1. How do linear and exponential growth models differ in predicting long-term behavior?
2. How can you graph both models to compare their fit to the data?
3. How would changes in $k$ affect the exponential growth rate?
4. What are some limitations of using a linear model for predicting revenue?
5. How can you determine which model better fits the data over a longer period?

Tip:

When choosing between linear and exponential models, always consider how the data behaves over time—exponential models often provide a better fit for rapidly growing trends.