Question

the table below shows retail sales at drug stores in the united states in billions of dollars since 2000.

year	retail sales
3	108.426
6	141.781
9	169.256
12	202.297
15	222.266

the graph shows a scatterplot of the data and the linear model ( y = 9.44x + 84.182 ), where ( y ) is billions of dollars and ( x ) is years since 2000.

a) is the line a good model for the data?

(\bigcirc) the line is a good model for the data.
(\bigcirc) the line is not a good model for the data

b) use the graph of the model to estimate what retails sales were in 2010.

(square) billions of dollars

c) use the equation of the model to predict the year when retails sales will reach $244 billion.

(square)

Explanation:

Part A

The scatterplot points lie close to the linear model line, showing a strong linear trend. So the line is a good model.

Step1: Determine x for 2010

2010 is 10 years since 2000, so \( x = 10 \).

Step2: Estimate from graph

From the graph, at \( x = 10 \), the y - value (retail sales) is around 180 (or using the equation \( y=9.44(10)+84.182 = 94.4+84.182=178.582\approx180 \)).

Step1: Set up the equation

We know \( y = 244 \) and the model \( y=9.44x + 84.182 \). So \( 244=9.44x + 84.182 \).

Step2: Solve for x

Subtract 84.182 from both sides: \( 244 - 84.182=9.44x \). So \( 159.818 = 9.44x \). Then \( x=\frac{159.818}{9.44}\approx16.93 \approx 17 \).

Step3: Find the year

Since \( x \) is years since 2000, the year is \( 2000 + 17=2017 \).

Answer:

The line is a good model for the data.

Part B