Question

Suppose a railroad rail is 6 kilometers and it expands on a hot day by 10 centimeters in length. Approximately how many meters would the center of the rail rise above the​ ground?

Answer 1

Given that the railroad rail is 6 kilometers before expansion.

When the railroad expands, it makes an isosceles triangle at the center of the rail.

The base of the triangle is 6 kilometers while each leg of the triangle is given by (6 + 10/100,000) / 2 = 6.0001 / 2 = 3.00005 kilometers.
[recall that 100,000 centimeters gives 1 kilometers, thus 10 centimetres gives 10/100,000 kilometers]

Taking one half of the isoceles triangle, we have a right triangle with one of the leg as 6 / 2 = 3 kilometers and the hypothenus = 3.00005 kilometers.

Thus, using the Pythagoras theorem, we find the other leg (height) of the right triangle as follows.

Let the other leg of the right triangle be h, then
$3^{2} + h^{2} = 3.00005^{2} \\ \\ \Rightarrow h^2=3.00005^{2}-3^2 \\ \\ =9.0003000025-9=0.0003000025 \\ \\ \Rightarrow h= \sqrt{0.0003000025} =0.01732058$

Thus, the center of the rail will rise 0.01732058 kilometers above the ground.

But, there are 1,000 meters in 1 kilometers, therefore, the center of the rail will rise 0.01732058 x 1,000 = 17.32 meters above the ground.