Equation Relation

Question

The average cost of a 30-second advertising slot during the Super Bowl increased linearly from $2.5 million in 2010 to $3.5 million in 2012. Find the equation that relates the cost (in millions of dollars) of a 30-second slot, y, to the number of years after 2010, x. What was the average cost in 2011? 

Answer 1

We can use the information to construct an equation of the line.

 

Since we are using year 2010 as the starting point, lets let x is to the number of years after 2010, so in year 2010, x has a value of 0.  Since in that year the cost was 2.5 million, we will let (0, 2.5) be that point ... y is in millions of dollars

 

For year 2012, x will be 2 (2 years after 2010) and y will be 3.5 (y in millions of dollars).  The coordinates of that point are (2, 3.5)

 

Using the slope formula, we can determine the slope of the line.

 

     (0, 2.5)    (2, 3.5)

      x₁   y₁     x₂   y₂

 

     m = (y₂ - y1)/(x₂ - x₁)

 

         = (3.5 - 2.5)/(2 - 0)

         = 1/2

         = .5

 

The slope of the line is .5.

 

Since point (0, 2.5) is on the line, and x = 0, the y-intercept is 2.5

 

Using the slope-intercept form,

 

    y = mx + b

 

    y = .1x + 2.5 is our equation .... y is in millions of dollars

 

To determine the price in 2011, substitute x = 1 ... (2011 is 1 year after 2010)

 

    y = .5(1) + 2.5

       = .5 + 2.5

       = 3

 

in 2011, the cost was 3 million