pythegorean theorum and distance?

Question

This is the problem: Train tracks are made of metal. They expand when the weather is hot and contract when it's cold. Normally tracks are set down with gaps between, giving them room to expand and contract. Consider one mile of track that is laid in one piece, nailed solidly to the ground at both ends, with no gaps. This track will expand by two feet in the heat and it will buckle in the middle. Use what you know about right triangles to determine how far off the ground the middle of the track will be.

My calculations give me: 145.341 ft. Is this right or wrong? (if wrong point me in the right direction please!!)

Answer 1

72.6705 feet. When the expansion occurs you will have two right triangles, each with a base of 1/2 mile (2640 ft). The hypotenus of the triangle will be exactly 1 foot longer (2641 feet). When solving for the third side you get the answer.

Answer 2

145 feet sounds a little high, doesn't it

I think this problem assumes that the track will buckle such that it approximates a tent shape with a high point in the middle sloping evenly back to both ends. Sort of like a real flat "A" without the cross piece.

Two right triangles are an approximation of the track after it buckles.

Now, the bottom, or adjacent leg of each right triangle is going to be half a mile long (the track started out one mile long).

Since the track expands by two feet, the new expanded length of the track, which forms the hypoteneus of each of the right triangles will be half-a-mile + 1 foot (1 foot of the 2 foot expansion going to each triangle).

now we now that a^2+b^2=c^2

we want to know the "opposite" angle which will be the "point" of the bent rail (we can call the bottom or adjacent "a" and the height or opposite "b") and remembering 1 mile = 5280 ft and so half a mile =2640 ft

that means that 2640^2+b^2=(2640+1)^2

a little algebra gives us:
b^2=(2641)^2-(2640)^2
b^2=6974881-6969600
b^2=5281
b=72.7

over 70 feet
that number surprises me but I think it is correct
better check my algebra and my arithmetic though

good luck

Answer 3

I get half your answer because I am assuming that the track being fastened on both ends would make a triangle raised in the middle. Each hypotenuse would be 5280/2 + 1 (only 1 foot longer than 1/2 mile). Your technique sounds right.

Answer 4

Look wrong, kemo sabe.
What you have is an isoceles triangle with two sides 2641 feet and a base of 5280 feet. So you can look at one-half as a right triangle of side 2640 feet and hypotenuse of 2641. The square of the other side (whose root is your answer) is 5281. The root looks like about 73. Apparently you missed cutting something in half.