Chapter 5.2, Problem 136AYU

Projectile Distance An object is fired at an angle $\theta$ to the horizontal with an initial speed of ${\nu }_0$ feet per second. Ignoring air resistance, the length of the projectile's path is given by

$L\left(\theta \right)=\frac{{\nu }_0{}^2}{32}\left[\sin \theta -{\left(\cos \theta \right)}^2\cdot \left(\ln \left[\tan \left(\frac{\pi -2\theta }{4}\right)\right]\right)\right]$

Where $0<\theta <\frac{\pi }{2}$.

Find the length of the object's path for angles $\theta =\frac{\pi }{6},\frac{\pi }{4}$ and $\frac{\pi }{3}$ if the initial velocity is $128$ feet per second.

Using a graphing utility, determine the angle required for the object to have a path length of $550$ feet if the initial velocity is $128$ feet per second

What angle will result in the longest path? How does this angle compare to the angle that results in the longest range? (Set Problems $121-124$.)