1-A rifle can shoot a projectile with a velocity of 207m/s. At what angle should the rifle be pointed to give the maximum range?
2-Evaluate the maximum range
The Attempt at a Solution
I considered at first that the problem seemed like it could be solved as an optimization problem. Since velocity-x is rcos(n), I used used x(t)=207cos(n)t to model the distance. I took the first derivative and got
v(t)=207cos(n)-tsin(n). I then realized that I couldn't set the derivative equal to zero and solve for t, because there was still (n) to deal with.
The book I'm using(3,000 solved problems in physics; schaum's) recommended to consider that the vertical displacement is zero, and solve for t in the equation x=v_0 + 1/2(a)t^2, giving t=x/v_0. Then it substituted that in for t in the same kinematic equation, but for the y-displacement. It then solved for x, used some trig identities, and then used a value provided in the book from a different problem(bad typo perhaps?).
Is there a much simpler way to solve this problem, or is the book's method the only way? Because it seems that the problem didn't give enough information, so it used another problem to help solve it; very confusing. It seems that calculus could be used for this, but I'm not sure.