Question

(6, 180.6)
(7, 182.9)
(8, 184.5)
(9, 185.7)
(10, 186.7)
(11, 187.5)
​(a) Find a linear model for the data with x=6 corresponding to the year 2006.
​(b) Assuming the trend​ continues, estimate the total sales for the year 2016.

Answer 1

(a) To find a linear model for the data, we need to find the equation of a line that best fits the given points. We can use the formula for the equation of a line, which is given by:

y = mx + b

where m is the slope of the line and b is the y-intercept.

To find the slope, we can use the formula:

m = (y2 - y1) / (x2 - x1)

Using the points (6, 180.6) and (11, 187.5), we have:

m = (187.5 - 180.6) / (11 - 6)
m = 6.9 / 5
m = 1.38

Now, we can substitute one of the points into the equation y = mx + b to solve for b. Using the point (6, 180.6), we have:

180.6 = 1.38(6) + b
180.6 = 8.28 + b
b = 180.6 - 8.28
b = 172.32

Therefore, the linear model for the data is:

y = 1.38x + 172.32

(b) To estimate the total sales for the year 2016, we can substitute x = 2016 into the linear model equation:

y = 1.38(2016) + 172.32
y ≈ 2787.68 + 172.32
y ≈ 2960

Therefore, the estimated total sales for the year 2016 is approximately 2960.