Use the expression in Problem 54 to find 

(a) The maximum range of a projectile with launch speed v_i and 

(b) The launch angle θ at which the maximum range occurs.

Data from in Problem 54

The range R of a projectile is defined as the magnitude of the horizontal displacement of the projectile when it returns to its original altitude. In other words, the range is the distance between the launch point and the impact point on flat ground. A projectile is launched at t = 0 with initial speed vi at an angle θ above the horizontal. 

(a) Find the time t at which the projectile returns to its original altitude. 

(b) Show that the range is