Sales The Coca-Cola Company had sales of 19,564 million in...

Sales The Coca-Cola Company had sales of $\$ 19,564$
million in 2002 and $\$ 35,123$ million in $2010 .$ Use the
Midpoint Formula to estimate the sales in $2006 .$
Assume that the sales followed a linear pattern.
(Source: The Coca-Cola Company)

Key Concepts

Midpoint Formula

The Midpoint Formula is a method used to find the central value between two data points. It involves taking the average of the two values, which assumes that the change between them is uniform. In the context of linear change, it provides a straightforward way to estimate an intermediate value.

Linear Interpolation

Linear interpolation is a technique used to estimate the value of a function within the range of two known points under the assumption that the function changes at a constant rate between these points. It simplifies complex data by presuming a straight-line relationship, making it a practical method for calculations in time series and other scenarios.

Assumption of Linearity

The assumption of linearity means that the relationship between the variables is considered to be a straight line, with a constant rate of change. This premise underpins methods like the midpoint formula and linear interpolation, allowing for simple and direct calculations without the need for more complex modeling.

Transcript

00:03 We were asked to answer a question about sales of a particular company.

00:10 So, this company, we're told, has sales of 19 ,564 million in 2002, and $35 ,123 million in 2010.

00:28 We're asked to estimate the sales in 2006 using the midpoint formula.

00:36 Additionally, we're asked to assume that the sales followed a linear pattern.

00:46 So first of all, let's identify these pairs of years and the sales amount.

01:00 Let's pair those and plot them on xy planes.

01:04 So we have sales of 19 ,564 in 2002.

01:09 So we've got, let's say, year and then sales amount.

01:16 So we have 2002 and 19 ,564.

01:29 And the other point was 35 ,123 million in 2010.

01:49 So let's just plot these two points.

01:52 To do this, notice that the x components lie much higher than zero.

02:01 As to the y components.

02:02 So i'm actually just going to draw a section of the first quadrant here.

02:08 So our x -axis, and to do this, i'm going to use a special notation.

02:13 So we have the origin where the x and y -axis intersect, but then we're going to draw this jagged line to indicate that we're skipping a lot of x values until we get up to 2002 and then all the way up to 2010.

02:31 Likewise for the y -axis, i'll draw the jagged line.

02:36 And then we'll start out at, let's say, 19 ,000, and we'll go up to 36 ,000 on the y -axis.

03:00 So first i'll plot point a.

03:03 So point a is located at 2002, 19 ,564.

03:08 This is slightly greater than 19 ,000.

03:12 Okay.

03:13 And then point b is located at 2010, 35 ,123...

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