Solve Slope Applications

At the beginning of this section, we said there are many applications of slope in the real world. Let's look at a few now.

Example 4.38

The 'pitch' of a building's roof is the slope of the roof. Knowing the pitch is important in climates where there is heavy snowfall. If the roof is too flat, the weight of the snow may cause it to collapse. What is the slope of the roof shown?

This figure shows a house with a sloped roof. The roof on one half of the building is labeled "pitch of the roof". There is a

Solution
Use the slope formula. \(m=\frac{\text { rise }}{\text { run }}\)
Substitute the values for rise and run. \(m=\frac{9}{18}\)
Simplify. \(m=\frac{1}{2}\)
The slope of the roof is \(\frac{1}{2}\).  
  The roof rises \(1\) foot for every \(2\) feet of horizontal run.

Table 4.40

Try It 4.75

Use Example 4.38, substituting the rise = 14 and run = 24.

Try It 4.76

Use Example 4.38, substituting rise = 15 and run = 36.

Example 4.39

Have you ever thought about the sewage pipes going from your house to the street? They must slope down \(\frac{1}{4}\) inch per foot in order to drain properly. What is the required slope?

This figure is a right triangle. One leg is negative one quarter inch and the other leg is one foot.

Solution
Use the slope formula. \( \begin{align} \begin{array}{c} m=\frac{\text { rise }}{\text { run }} \\ m=\frac{-\frac{1}{4} \text { inch }}{1 \text { foot }} \\ m=\frac{-\frac{1}{4} \text { inch }}{12 \text { inches }} \end{array} \end{align} \)
Simplify. \(m=-\frac{1}{48}\)
  The slope of the pipe is \(-\frac{1}{48}\).


The pipe drops 1 inch for every 48 inches of horizontal run.

Try It 4.77

Find the slope of a pipe that slopes down \(\frac{1}{3}\) inch per foot.

Try It 4.78

Find the slope of a pipe that slopes down \(\frac{3}{4}\) inch per yard.