SOLUTION: A strip of metal 2.5km long is clamped to the ground during the winter with one clamp at each end. It expands 1m in length during the hot summer, bulging up in the middle as a resu
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Question 1133887: A strip of metal 2.5km long is clamped to the ground during the winter with one clamp at each end. It expands 1m in length during the hot summer, bulging up in the middle as a result. What is the highest distance, in m, that the metal strip could be above ground in the middle? Answer by greenestamps(13198) (Show Source):
In order to answer the question, we need to assume that the bowed metal strip is an arc of a circle. Here is a nice picture, along with a brief discussion of how to find the radius of the circle from the arc length and chord length. Once we know the radius of the circle, we can find the answer to the given question.
Using a slight modification of the diagram shown on that page, let c/2 be half the chord length and a/2 be half the arc length; and let x be the measure of central angle BCA. So a/2 = 2501/2 and c/2 = 1250.
(1) Using radian measure, the arc length a/2 is the radius of the circle times x:
(2) The sine of angle BCA (opposite over hypotenuse) is 1250/r:
Solving both equations for x gives us
(1)
(2)
So we have an equation we can solve to find the radius r:
There is no algebraic method for solving this equation. I used my graphing calculator to find the radius to be 25528.3m.
Note: I also verified this result using wolframalpha.com. If you are not familiar with that web site, you should become familiar with it. It can do amazing things. I went to the web site and entered "find radius of circle if chord length is 2500 and arc length is 2501"; it interpreted the question correctly and gave the answer within a few seconds.
Once we have the radius of the circle, we can proceed to the answer to the question.
= 0.048985 radians (to 6 decimal places)
The height we are looking for in the problem is the difference between the radius of the circle and the length of BC:
The surprising answer to the question is that the middle of the metal strip will be about 30.6m above the ground.
