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This table indicates that range increases as theta increases, up to Pi/4, and then begins to decrease. The following graph, which plots range vs. angle, confirms that Pi/4, 45 degrees, does produce the maximum range. Again range function is produced by determining the x coordinate for the time t (Solve[ y[t,theta]==0,t]) at which the projectile strikes the ground.

Input := 


range[theta_] =  x[t,theta]/.

	Solve[ y[t,theta]==0,t] [[2]];

Input := 


Plot[range[theta], {theta,0,Pi/2} ]

Output =


-Graphics-