Please send me a help. Please show all your work and highlight the answers. Thank you!

P.S. I also put example 6. You could use it to solve the problem!

Problem 5: The trajectory of a roller coaster can be described by the following parametric equation F(0) = x(0)i + y(0)j + =(0)k where x(0) = (a + bsin @) cose Y(0) = (a + b sin @) sin e z(0) = b+ bcose a and b are constants. If e is a function of time, i.e. @ (t), then the parametric equation described certain motion for a moving point along the trajectory. Let @ (t) = wt where w is constant. Find the tangent vector for this curve at any point on the curve, i.e. find the velocity of this moving dr point, v = = =? Hint: refer to example 6 in lecture $ notes. Example 6: A roller coaster based on the parametric equations of a 3D Lissajous curve r(t) = sin(s(t)) i + sin(2s(t)) j + cos(s(t)) k s(t) represents the distance along the track. Obtain the velocity of the roller coaster. Solution: x(t) = sin(s(t)) ; y(t) = sin(2s(t)) ; z(t) = cos(s(t)) dx(t) dt - = cos(s(t))- = cos(s(t)) s dy(t) dt -= 2 cos(2s(t)) - =2cos(25(t)) s dz(t) ds dt = - sin(s(t)) it = - sin(s(t)) s The velocity is then dr(t) D(t) = dt = (cos(s(t)) i + 2 cos(2s(t))] - sin(s(t)) k)s