The table gives the approximate economic value associated with various levels of oil recovery in a U.S. state.
Percent Recovery (%)	10	40	50	80
Economic Value ($ billions)	500	1,200	1,300	2,300
Find the regression line.
d(r) = 
 
Use the regression line to estimate the economic value (in billions of dollars) associated with a recovery level of 70%.
$ 
 billion

To find the regression line, we need to determine the linear relationship between the percent recovery ($ r $) and the economic value ($ d $). This involves calculating the slope ($ m $) and intercept ($ b $) of the linear equation, which is of the form:

\[
d(r) = mr + b
\]

The data points given are:
- $ (r, d) = (10, 500), (40, 1200), (50, 1300), (80, 2300) $.

### Step 1: Calculate the slope $ m $
The formula for the slope is:

\[
m = \frac{n \sum (r_i d_i) - \sum r_i \sum d_i}{n \sum r_i^2 - (\sum r_i)^2}
\]

Where:
- $ n $ is the number of data points (here $ n = 4 $),
- $ r_i $ and $ d_i $ are the respective percent recovery and economic value pairs.

### Step 2: Calculate the intercept $ b $
Once we have the slope, the intercept can be calculated using the formula:

\[
b = \frac{\sum d_i - m \sum r_i}{n}
\]

After we find the equation $ d(r) $, we'll use it to estimate the economic value for a 70% recovery rate.

Let's perform the calculations.The regression line based on the data is:

\[
d(r) = 25.4r + 182
\]

Using this equation, the estimated economic value for a recovery level of 70% is:

\[
d(70) = 25.4(70) + 182 = 1960 \, \text{billion dollars}
\]

Would you like more details or have any further questions?

Here are five related questions for further exploration:
1. How do we interpret the slope of the regression line in this context?
2. What would the economic value be for a 90% recovery using the same regression line?
3. How does the regression line compare with the actual data points?
4. What is the economic value when the percent recovery is 0%, according to the regression line?
5. Can this linear regression model be used for any percent recovery outside the given data range?

**Tip:** Always check the residuals (differences between actual and predicted values) to assess how well a regression line fits the data.

The table gives the approximate economic value associated with various levels of oil recovery in a U.S. state. Percent Recovery (%) 10 40 50 80 Economic Value ($ billions) 500 1,200 1,300 2,300 Find the regression line. Use the regression line to estimate the economic value (in billions of dollars) associated with a recovery level of 70%.

Learn how to calculate the regression line for oil recovery data, finding the economic value associated with various percent recoveries. This detailed solution includes key algebraic concepts, such as the slope and intercept formulas, and estimates the economic value for a 70% recovery rate. Suitable for students in Grades 11-12.

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