Let's go step by step through the questions from the image:

a. Find the company's equation:

You are asked to make a linear equation using the 2002 price ($34.99) and the 2005 price ($43.43) for the company’s model.

1. The formula for a line is:
   $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept.

2. The slope $m$ can be calculated as: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Using 2002 as the starting year (0 years) and 2005 as the 3rd year: $m = \frac{43.43 - 34.99}{3 - 0} = \frac{8.44}{3} = 2.8133$

3. The equation for the company using 2002 as the base year would be: $Pc(t) = 2.81t + 34.99$

b. Find the consumer advocate’s equation:

Now, using the 2002 price ($34.99) and the 2009 price ($55.37), we apply the same method to find the slope.

1. The slope $m$ is: $m = \frac{55.37 - 34.99}{7} = \frac{20.38}{7} = 2.9114$

2. The equation for the consumer advocate becomes: $Pa(t) = 2.91t + 34.99$

c. Predictions for 2015 (t = 13):

Substituting $t = 13$ into the company’s and advocate's equations:

1. Company's projection: $Pc(13) = 2.81(13) + 34.99 = 36.53 + 34.99 = 71.52$

2. Consumer advocate’s projection: $Pa(13) = 2.91(13) + 34.99 = 37.83 + 34.99 = 72.82$

d. Reason for choosing those points:

Both the company and consumer advocate used points that capture the trend of rising prices. The company chose an earlier time (2002-2005) to model a potentially lower rate of increase, while the consumer advocate used a longer timeframe (2002-2009), including more data and possibly identifying a steeper trend.

e. Relative increase from 2002 to 2010:

The price increased from $34.99 in 2002 to $61.99 in 2010.

1. The relative increase is: $\frac{61.99 - 34.99}{34.99} \times 100 = \frac{27}{34.99} \times 100 \approx 77.18\%$

The closest estimate would be 75-85%.

Let me know if you'd like any further clarification! Here are five related questions you might find helpful:

1. How do we interpret the slope in these types of equations?
2. What would happen if we included more years in the model?
3. How would the predictions differ if we used non-linear models?
4. Could inflation have been considered in these models?
5. How do consumer advocates typically assess price trends?

Tip: Always check the units and timeframes when creating models to ensure your predictions align with real-world changes.