Question 1111614: Train tracks are made of metal. Consequently, they expand when it’s warm and shrink when it’s cold. When riding in a train, you hear the clickety-clack of the wheels going over small gaps left in the tracks to allow for this expansion. Suppose you were a beginner at laying railroad tracks and forgot to put in the gaps. Instead, you made a track 1 mile long that was firmly fixed at each end. On a hot day, suppose the track expanded by 2 feet and therefore buckled up in the middle, creating a triangle.
Roughly how high would the midpoint be?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! track is 1 mile long which is equal to 5280 feet.
on a hot day, the track expands 2 feet.
the track buckles and forma a triangle.
the height of this triangle would be the altitude of an isosceles triangle whose base is 5280 feet in length and whose two other sides would be 5282 / 2 = 2641 feet each.
if you drop a perdendicular from the peak height of this triangle vertical to the base, then you form 2 right triangle, each with one base leg of 2640 feet and a hypotenuse of 2641 feet.
you can use the pythagorus formula to find the height of the right triangles.
pythagorus formula states hypotenuse squared = first leg squared plus second leg squared.
your hypotenuse is 2641 feet in length
one of the legs is 2640 feet in length
let the other leg be x feet in length
the formula becomes 2641^2 = x^2 + 2640^2
solve for x^2 to get x^2 = 2641^2 - 2640^2
solve for x to get x = plus or minus square root of (2641^2 - 2640^2)
minus is no good since the measurement has to be positive, so you get:
x = square root of (2641^2 - 2640^2).
you get x = square root of 5281.
that makes x approximately equal to 72.6704892 feet.
that's the distance between the original position of the steel rail and the peak of the buckle caused by the expansion of the steel.
my diagram looks like this:
the length of the original track is equal to AC.
the expanded length of the original track is equal to AB and BC.
the right triangles formed are ABD and CBD
these triangles are congruent, so we only need to use one of them to find the length of the distance between the top of the buckle and the position of the original track.
that would be the length of BD.
BD is perpendicular to AC, therefore triangles ABD and CBD are right triangles.
use pythagorus formula to get length of BD.
pythagorus says:
[BD]^2 + 2640^2 = 2641^2
solve for [BD]^2 to get:
{BD}^2 = 2641^2 - 2640^2
that's what leads to the previously established value of BD equal to 72.6704892 feet.
that's the distance between the position of the original rail before expansion and the position of the peak of the buckle of the rail after expansion.
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