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Railway going up a mountain.
Steep scenic railway in the Blue Mountains, Australia.

Slopes of lines — as well as pyramids, railways and roofs

The slope of a line is a precise mathematical formulation of how steep it is. This is a concept that has important ramifications when building roads, railways, houses and ski jumps. Even ancient pyramids would have been built with a firm eye on how steep things are!

In this step we

  • solidify your mathematical understanding of the important notion of slope

  • look at a variety of examples of slopes in practice.

What is the slope of a line?

The slope of a line is a number \(\normalsize{m}\) which is defined as

\[\Large{\operatorname{slope} = m = {\text{change in y} \over \text{change in x}}.}\]

So the line \(\normalsize{AB}\) that goes through the points \(\normalsize{A=[2,1]}\) and \(\normalsize{B=[7,4]}\) has slope

\[\Large{m={4-1 \over 7-2} = {{3}\over{5}}}.\]

We often use the Greek symbol \(\Delta\) to denote a change, in which case we would write

\[\Large{\operatorname{slope}={\Delta y \over \Delta x}}.\]

Note that the slope of a vertical line is not defined, since in that case the denominator would be zero, which is not allowed. Another way of expressing this is that a vertical line has slope equal to “infinity”.

Picture of line AB with two points A,B and the triangle showing change in x and change in y

Because of similar triangles, if we choose two other points on the same line, the calculation of the slope would give the exact same value. So the slope really does depend on the line, and not on the two particular points we choose on it.

Q1 (E): What is the slope of the line \(\normalsize{CD}\) where \(\normalsize{C=[-2,5]}\) and \(\normalsize{D=[4,-3]}\)?

The connection with direct proportions

The simple line \(\normalsize{y=mx}\) represents a direct proportionality with constant \(\normalsize{m}\). For each increase of \(\normalsize{1}\) in the \(\normalsize{x}\) value, \(\normalsize{y}\) increases by \(\normalsize{m}\). This number \(\normalsize{m}\) could also be \(\normalsize{0}\), in which case \(\normalsize{y}\) doesn’t change at all, or could be negative, in which case \(\normalsize{y}\) actually decreases as \(\normalsize{x}\) increases.

When we visualize \(\normalsize{y=mx}\) as a line in the Cartesian plane, we see that \(\normalsize{m}\) is exactly the slope of this line as defined above. In the following figure we see lines together with their various slopes.

Some lines through the origin and their slopes, m=1,2,3,-1,-5/2

Q2 (C): If the line \(\normalsize{y=mx}\) is perpendicular to the line \(\normalsize{y=nx}\), then what is the relation between \(\normalsize{n}\) and \(\normalsize{m}\)?

Slope or pitch of roofs

Slope is an important concept in the design and construction of roofs: and is in that industry expressed as the ratio of the vertical rise to the horizontal run, and usually called the pitch of the roof. A roof that rises 4 inches for every 1 foot or 12 inches of run is said to have a slope or pitch of “\(\normalsize{4}\) in \(\normalsize{12}\)” or “\(\normalsize{4:12}\)”. This notation reinforces the idea that it is the proportion between the two quantities that matters, not the absolute values themselves. A pitch of \(\normalsize{4:12}\) is the same as a pitch of \(\normalsize{2:6}\), or \(\normalsize{1:3}\).

While steep roofs are more difficult to build, and more dangerous to work on and fix, they have advantages. Thatch roofs for example need to be steep to allow water to drain from them effectively. Roofs in areas where there is a lot of snow need to have a high pitch so that the snow does not accumulate over winter.

Picture of thatch roof© “Anne Hathaway’s cottage” by Michael Zawadski/Wikimedia Commons CC BY SA 2.0

Gradients or slopes of railways

Slopes of lines are important for railway engineers. On a \(\normalsize{1\%}\) (1 in 100) gradient, a locomotive can pull half (or less) of the load that it can pull on level track. Modern railways have improved technology, and in fact the steepest part of the Lisbon tram system, located in Calçada de São Francisco, has a record gradient of 14.5% (about 1 in 6.9).

Of course for railways with systems of cables or interlocking cogs, the gradient can be much higher. For example, Pilatus railway (Switzerland) at 48% and the steepest in the world Scenic railway (Australia) at 52%.

Gradients or slopes of streets

The residential street with the steepest gradient is Baldwin street in Dunedin, New Zealand, with a gradient of about \(\normalsize{35\%}\).

Picture of steepest streetA House on Baldwin Street” Public Domain/Wikimedia Commons

While gradient is usually measured as a percentage, the corresponding slope would be just a number, so that a \(\normalsize{30\%}\) gradient would correspond to a slope of \(\normalsize{\frac{30}{100}}\) or \(\normalsize{0.3}\). A gradient of \({\normalsize 100 \% }\) corresponds to a slope of \({\normalsize 1}\), with an angle of inclination of \({\normalsize 45}\) degrees. Here is a picture that shows lines with various gradients.

Graph showing lines with gradients of 5%, 10%, 20%, 30%, 40%

Q3 (M): What do you guess is the maximum average gradient of a Tour de France leg?

Q4 (M): How about the slope for the Mayan pyramid at Chichen Itza?

Chichen Itza step pyramid© “Chichen Itza pyramid” by Att309/Wikimedia Commons CC BY SA 3.0

Answers

A1. The slope is \(\normalsize{s=(-3-5)/(4--2)=-8/6=-4/3}\).

A2. Perpendicular lines have slopes which multiply to \(\normalsize{-1}\). This is a lovely fact. One way to see it is to consider just lines through the origin \(\normalsize{O=[0,0]}\). If \(\normalsize{A=[a,b]}\) is another point, then the line \(\normalsize{OA}\) will have slope \(\normalsize{m={{b-0} \over {a-0}} = {b \over a}}\). How about a line perpendicular to \(\normalsize{OA}\)? That is the line \(\normalsize{OB}\) where \(\normalsize{B=[-b,a]}\), and it will have slope \(\normalsize{n=a/(-b)}\). So the product \(\normalsize{nm}\) will be \(\normalsize{-1}\).

Here is a picture which illustrates these perpendicular lines in the case of \(\normalsize{A=[3,5]}\).

Picture of two perpendicular lines through the origin, one through  [3,5], the other through [-5,3]

A3. On the Tour de France route for 2015, the highest average gradient (10.1%) was in Stage 14: Rodez to Mende.

A4. El Castillo, the main pyramid at Chichen Itza is 55.3 metres wide by 30 metres tall, resulting in a slope of just over \(\normalsize 1\).

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