Chapter 2.3, Problem 16E

A trough of length L has a cross section in the shape of a semicircle with radius r. (See the accompanying figure.) When filled with water to within a distance h of the top, the volume, V of water is $V=L\left[0.5\pi r^2-r^2\arcsin \left(\frac{h}{r}\right)-h{\left(r^2-h^2\right)}^{1/2}\right]$

Suppose $L=10\text{ft},r=1\text{ft},$ and $V=12.4{\text{ft}}^3.$ Find the depth of water in the trough to within $0.01\text{ft}$.