   SEARCH HOME Math Central Quandaries & 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.

Penny      Math Central is supported by the University of Regina and the Imperial Oil Foundation.