10/10/2023Kindly help with the below problem

F(x, y, z, a) := (x cos[a] - y sin[a], y cos[a] + x sin[a], z) where x, y and z are the positions of the three machine axes (in inches) and a is the angle of the rotary (here holding the material on the diagonal of the machine). 1. The machine is controlled using a language called GCode, and for an instruction moves each axis at a constant rate between two values. As an example moving from (0, 0, 0, 0) to (10, 10, 10, x/2) (lengths units in inches and radians for the angle), would be given by 10 10 10 f(t) = (-t, -t, -t, - t), with t E [0, s]. Where s is the time taken for the move, and f(t) = (x, y, z, a) is used to control the machine. Use the chain rule to find the velocity of the machine as a function of time. 2. For this machine with a given tool and material the maximum speed is 120 IPM (Inches Per Minute), how might you find s so the machine never goes faster than this? To do this you might find you have a rather complex expression, but it does actually simplify. You may just describe the process or use computer help to find the simpler form, only the brave or foolish should try by hand