the pitch of a roof is the number of feet the roof rises for each 12 feet horizontally.

Question

If a roof has a pitch of 8, what is its slope expressed as a positive number? How can I write this out as an equation?

Answer 1

If the pitch is 8, it means the roof raises 8 ft. per each horizontal 12 ft. 
Since slope, m, is defined as the rise/run; then m = 8/12.  

Simplifying, we obtain a slope m = 2/3

In order to write the line of the roof as an equation, let's assume that the roof line starts at the origin (0, 0); it rises 8 units for every 12 it advances; up to the point (12, 8).  At this point, the roof descends at the same rate until it reaches the point (0, 24).  We need to set two equations with the given constraints:

• y = mx + b  since the roof ascend from (0, 0) up until (8, 12) we use a positive slope:

         

          y = 2/3x + b  If we use (0, 0) to define b, then the equation of this section of the roof will be:
        

          y = 2/3x     ⟨0 ≤ x ≥ 12⟩

 

• At (12, 8) the roof starts descending at the same rate, then we use the negative slope m = -2/3:

     

         y = -2/3x + b  Now, let's use (12, 8) to define b: 

 

          8 = (-2/3)(12) + b   Simplifying, we obtain b = 16, ∴ the equation of this section of the roof will be:

 

          y = -2/3x + 16     ⟨12 ≤ x ≤ 24⟩