Math Question - Answer Seems Unreal !!

[kuiperbelt]

A perfectly horizontal metal handrail constructed across the USA between Baltimore, Maryland and San Francisco in California measures 2,580 miles long. The chief engineer on the project informs you that in the summer the handrail will expand by just 1 inch (1") and will buckle upwards. Find how far a pedestrian has to reach up to stay in contact with the handrail at the half way point. Important Note: Ignore the curvature of the earth.

[Tilly2]

I don't understand the reach up bit.

[Jonathan-Joe]

The handrail would expand by 1 inch if free to do so. For this problem we assume the two ends are fixed so the "expansion" makes the rail buckle upwards.

[JJ109]

753.3923264 feet approx

If I made a mistake, I'm not doing it again!

[bhg481]

This is a variation of the rope-around-the-earth problem. If a rope is laid around the equator (approx 25,000 miles) how much longer would it need to be to have a 1" gap all round?

[JJ109]

When it buckles an isosceles triangle is formed. You need the height at the midpoint.
The distance between Baltimore and the midpoint = 1290 miles
The hypotenuse of the triangle Baltimore, Midpoint, Height = 1290 miles andhalf an inch
Half an inch = 1/12672 miles
Square the hypotenuse and subtract the square of the base
Square root the remainder
and that = my answer!

[JJ109]

^^ half an inch - 1/126720 miles

[Gizmonster]

//// 753.3923264 feet approx

If I made a mistake, I'm not doing it again! ////

I didn't believe it, so I worked it out myself and got the same answer:

Same as JJ109 but I converted to inches.
1 mile = 63 360 inches.
Halfway point = 1290 x 63 360 = 81 734 400"
Triangle formed with hypotenuse = 81 734 400.5"
and base = 81 734 400"
Height of hypotenuse triangle = H
H = Sq Root ((81 734 400.5)^2 - (81 734 400)^2)
H = Sq Root (81 700 000)
H = 9038.88"
H = 753.23 feet

Wowzer !!!!

[nescio]

Bhg481
That is a different situation entirely. The equally surprising answer to that one is that the rope only needs to be 6.28” longer.

[bhg481]

nescio yes, otherwise known as 2 pi

[bobbinwales]

Doesn't it mean thw whole handrail increases by 1 inch...so won't extra height be 1 inch divide by 2 pi??
Maybe I misunderstand the question??

[ClareTG0ld]

I suppose the point is that when the handrail "buckles", its endpoints stay fixed, so it has to accommodate the extra inch of length by bending upwards. But that extra inch of length leads to a *lot* of extra height in the middle, as shown by the answers already given.

[bobbinwales]

OK so it's a triangle shape not an arc? So answer is maybe as jj109 etc said. Bhg this seems different to the rope one where whole rope is raised by 1 unit

[bhg481]

bob - both solutions are surprising at first thought; one is much larger and the other is much smaller than you expect.