I am having trouble solving part (e) of question 13 in the Kinematics section of the Appendix for AP Physics 1 Essentials. I got parts a-d, but must be missing something to solve for the height.

Thanks!

Part e is quite difficult (beyond scope of AP-1) and I highly recommend using a calculator or computer for numeric simulation.  You need to take your kinematic equations for horizontal and vertical and put them together into an equation with h as your variable.  The bad news -- there is no easy way to get this equation into an expression for just h, but rather, you may have something like h+Btanh+3h^2/sinh = C     That's not the equation, but you perhaps get the idea.  Then plug that into a computer or calculator to find a numeric solution.  Verify your solution works, and you're done.