 Now we're going to figure out the angle to get the maximum range. This derivation is the most complex, so hang in there. Our general strategy to find the maximum range is to get an expression for the range x max in terms of theta, and then figure out what value of theta will make x max the greatest. x max the range is just vix times t, the time of flight. Vix can be replaced with vi times cos theta, but what about t? Well, the time of flight can be found by setting the height y equals 0 and then solving for t. The right hand side can then be rewritten as. Then, since y equals 0, we can divide both sides by t and then rearrange the final equation to find that t equals 2 times vi times sin theta on g. Now we can put the time of flight t into our equation for x. We end up with x is equal to 2 times vi squared times cos theta times sin theta over g. So we want the value of the theta that makes x the biggest. It's not obvious what theta should be to make x as big as possible, but if we remember the trigonometry rules, we can rewrite 2 times sin theta times cos theta as sin of 2 theta. Making this replacement, the range simplifies down to x equals vi squared times sin of 2 theta over g. Since vi and g are fixed, the maximum x occurs when sin of 2 theta is maximized. The greatest value that sin can ever have is 1, which occurs when the argument is 90 degrees. This means that 2 theta is 90 degrees when sin is 1 and the range is maximized, which means that theta is equal to 45 degrees. So at 45 degrees, the range of the dolphins jump is the greatest it can be.