Math CentralQuandaries & Queries


Question from Tom:

Is there an algebraic means to determine the highest point of a parabolic arc if the base and perimeter are known?

Hi Tom,

A while ago we got a question from Jane where she had the base and height of a parabolic arc and wanted to know the length of the parabolic arc. The expression I used was

\[s = \sqrt{a^2 + 4 h^2} + \frac{a^2}{2 h} \sinh^{-1} \left(\frac{2h}{a} \right).\]

where $s$ is the length of the arc, $h$ is the height and $a$ is half the length of the base as in the diagram in my response to Jane. You can substitute the values you have for $s$ and $a$ in the expression above and you will be left with an equation containing $h$ as its only variable. Unfortunately you can't solve this equation for $h,$ the best you can do is to approximate a solution. You can use the Newton's Method or perhaps you have access to software that will perform the approximation.


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