[Math Lair] The Expanding Bridge

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Suppose that a bridge is a mile long, as shown. Let's assume that the bridge is level:

diagram of a 1 mile long bridge

When the weather gets hot, bridges, like everything else, tend to expand. Typically engineers designing a bridge would include small gaps over the span to allow for such expansion, but let's assume that they've forgotten about that in this case, and the bridge becomes four feet longer during a heat wave. Let's also assume that, instead of this expansion causing the bridge to crack and crumble, the expansion occurs such as to make the centre of the bridge rise, as shown below:

diagram of bridge after expansion

Note: The above diagram is not drawn to scale.


  1. What is the height of the bump (marked with a ? in the second diagram above)?
  2. (requires basic trigonometry) As a car travels towards the middle of the bridge, what is the angle of ascent?
[Postcard of the Williamsburg Bridge, New York.]


For question 1, we can use the Pythagorean theorem to find the height of the bridge. We can represent each half of the bridge as a right triangle with hypotenuse ½(5,280 + 4) = 2,642 feet, and one of its legs ½(5,280) = 2,640 feet. The other leg, which we'll represent by h, is the height of the bump in the centre of the bridge. So, we have:

h² + 2640² = 2642²
h² = 10564
h = 102.8 feet (approximately)

Surprising, isn't it? It seems more reasonable to expect that an expansion of four feet would result in a negligible bump, but that isn't the case.

For question 2:

tan θ = 102.8 ÷ 2640
θ = 2.23° (approximately)

So the bump itself isn't really all that steep. It would be noticeable for someone walking over the bridge, but driving over the bridge might not feel significantly different than driving over the bridge with no bump, except at the centre of the bridge.

Sources used (see bibliography page for titles corresponding to numbers): 44.