The following table shows retail sales in drug stores in billions of dollars in the U.S. for years since 1995.



YearRetail Sales

0

85.851

3

108.426

6

141.781

9

169.256

12

202.297

15

222.266


Let S(t)𝑆(𝑡) be the retails sales in billions of dollars in t years since 1995. A linear model for the data is F(t)=9.44t+84.182𝐹(𝑡)=9.44𝑡+84.182.Estimate the retails sales in the U. S. in 2013.  billions of dollars.
Use the model to predict the year in which retails sales will be $243 billion.

To solve the given problem using the linear model $ F(t) = 9.44t + 84.182 $, we will first estimate the retail sales in the U.S. in 2013 and then determine the year in which retail sales will be $243 billion.

### Estimating Retail Sales in 2013

1. **Determine $ t $ for the year 2013:**
   \[
   t = 2013 - 1995 = 18
   \]

2. **Substitute $ t = 18 $ into the linear model $ F(t) = 9.44t + 84.182 $:**
   \[
   F(18) = 9.44 \times 18 + 84.182
   \]

3. **Calculate the value:**
   \[
   F(18) = 169.92 + 84.182 = 254.102
   \]

So, the estimated retail sales in the U.S. in 2013 are **$254.102 billion**.

### Predicting the Year when Retail Sales will be $243 Billion

1. **Set $ F(t) = 243 $ and solve for $ t $:**
   \[
   243 = 9.44t + 84.182
   \]

2. **Isolate $ t $:**
   \[
   243 - 84.182 = 9.44t
   \]
   \[
   158.818 = 9.44t
   \]
   \[
   t = \frac{158.818}{9.44}
   \]
   \[
   t \approx 16.82
   \]

3. **Determine the year:**
   \[
   \text{Year} = 1995 + 16.82 \approx 1995 + 17 = 2012
   \]

So, the retail sales are predicted to be $243 billion around the year **2012**.

### Summary
- The estimated retail sales in the U.S. in 2013 are $ \$254.102 $ billion.
- The year in which retail sales are predicted to be $ \$243 $ billion is approximately 2012.

Would you like further details or have any questions?

### Related Questions

1. What is the formula for a linear model, and how is it used to make predictions?
2. How can we verify the accuracy of a linear model using given data points?
3. What are the limitations of using a linear model for long-term predictions?
4. How can we find the slope and y-intercept from a set of data points?
5. What other types of models could be used to predict retail sales, and how do they compare to linear models?
6. How can we use residuals to evaluate the fit of a linear model?
7. What factors might cause actual retail sales to deviate from the predictions of the linear model?
8. How would the model change if we included more recent data points?

### Tip
When working with linear models, always plot the data points and the fitted line to visually inspect the model's fit and identify any potential outliers or patterns that the model does not capture.

Learn how to estimate retail sales using linear models with a step-by-step solution. Predict retail sales in the U.S. for 2013 and determine the year when sales will be $243 billion using the provided linear model. Detailed calculations and explanations included.

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