The $100-$meter diameter wheel needs to be designed so that it rotates at a constant rate and every complete revolution takes $12$ minutes. The wheel has been installed in a vertical plane and the lowest seat is $2$m above the ground surface. Each ride consists of only one full rotation.$($a$)$ You are required to obtain an expression for the height of a rider above the ground in terms of both the angle $0$ radians the wheel rotates through and the time t$,$ assuming that the rider is initially at a seat that is $2$m above the ground surface at the start of the wheel's journey.$($b$)$ Find a general expression for the rate of ascend for this rider in terms of both the angle of rotation and time.$($c$)$ Determine how fast this rider is moving vertically when t $=560$ sec and state whether it isascending or descending. Justify your answer.$($d$)$ Find a further general expression for the horizontal displacement of the rider from its initial position with respect to both time and angle of rotation and hence determine the speed of the rider when t $=180$ sec$\mathrm{.}14($e$)$ Using Calculus, determine what the maximum horizontal and vertical speeds of the rider would be and at which points during the ride they occur.$($f$)$ State expressions for the acceleration of the rider in both horizontal and vertical directions and show that the resultant acceleration experienced by the rider is independent of where on the ride he$/$she might be$.(8)$ in desiening new whas pil the same diacons and sat i ride durin as before i is proposedde $=0\mathrm{.}00049365$ t$05$ in rad$/$sec during the first half of the ride when the rider is ascending and de $=0\mathrm{.}017895$ t$-0\mathrm{.}1882$ in rad$/$sec during the second half, when the rider is descending.Determine the total angular displacement of the wheel between the time intervals of t $=250$ secand t $=550$ sec$.$ Also calculate the total curved distance that the rider travels between the times oft $=200$ sec and t $=720$ sec$.($h$)$ To make the ride more exciting it is proposed that the wheel should ascend in such a way that therate of change of & with respect to t follows an exponential model of the form de $=$ aet during thefirst quarter of the ride and df $=$ be mt during the second quarter with a$,$ b$,$ m and k all beingconstants. The same patterns are followed during the second half of the ride. The wheel diameter and ride duration still unchanged.Suggest possible values for the constants mentioned and use them to evaluate the angle through which a Rider goes between the time intervals of t$=150$ sec and of t$=250$ sec