Given the equations of projectile motion (no air resistance), it is easy to find the launching angle theta that produces the maximum range. That angle is 45 degrees. The maximum time of flight is obtain instead for a launching angle theta=90 degrees: the projectile is launched straight up and the range is zero.
I would like to find the launch angle necessary to obtain the maximum range and maximum time of flight simultaneously. There must be a launch angle such that the obtained range and time of flight may not each be the maximum but together are the the largest. How would I set the problem up to find this special launch angle? Do I need express both time of flight and range as a function of theta,i.e. R(theta) and T(theta), multiply the two functions and set the derivative to zero? Should I find the maximum of the product R(theta)*T(theta) or the maximum of the sum R(theta)+T(theta)? Or something else?