Working on a calculus project, questions circled are not yet finished. Could use some help in solving. Thank you for your help!

Guided Project 22: Landing an airliner Topics and skills: Derivatives, Chain Rule, integration In this project we construct a model for the flight path and velocity profile of an airliner approaching a runway for landing. We assume that the airliner is flying horizontally at an altitude of & miles at a horizontal distance of L miles from the airport when it begins its descent. On landing we assume that the flight path is horizontal. 1. The coordinate system shown in Figure I ( which is not drawn to scale) places the airport at the origin and the point at which the descent begins is (-L, h). We aim to find a function yur) that gives the flight path of the airliner. Explain why the assumptions given above imply that M-L)=h. NO)=0. J(-L)=0, y'(0) =0. 2. Explain why a linear function or a quadratic function cannot satisfy all four conditions. Figure I 3. In light of Step 2. it makes sense to begin with a cubic polynomial. Assume that j(x) = ar' + by' + cr + d. Use the conditions in Step I to determine the coefficients a, b, c. and d. Show that 4. Verify that this function satisfies the conditions in Step 1. Assume that & = 100 miles and h = 6 miles (about 31,700 feet) and confirm that the graph in the figure is accurate. 5. Steps 1-4 give us the shape of the flight path, but say nothing about the speed of the airliner on that path. Begin by making the (unrealistic) assumption that the horizontal speed of the airliner is constant during the descent; that is, deidr = Ma. Use the Chain Rule in the form dy dy dx to show that the vertical velocity of the airliner along the path is X - SAY.(*)+). With up - 450 mi/hr, L = 100 miles, and h = 6 miles, graph the vertical velocity function. Where along the flight path is the magnitude of the vertical velocity a maximum? A minimum? 7. Use the Chain Rule to show that the vertical acceleration " of the airliner along the flight path is day _ 6head (2x + 1)Landing an airliner 43 8. With No - 450 mi/hr, L = 100 miles, and A - 6 miles, graph the vertical acceleration function. Where along the flight path is the magnitude of the vertical acceleration a maximum? A minimum? 9. Explain how the vertical velocity can be zero at the landing, while the vertical acceleration is a maximum. What does this say about the roughness of the landing? 10. A more realistic model assumes that the horizontal component of the velocity decreases as the airport is approached. Assume that the horizontal velocity decreases from to = 450 mi/hr to w = 150 mi/hr over the descent (-LS x 50) and that the descent takes 7 hours. One could use many different functions to describe the decreasing horizontal velocity. Let's try u(!) = use * (note that my(0) = My = 450, as specified). Use the condition M(7) = m = 150 to show that & = (In 3)/T. 1 1. Let x(r) be the horizontal distance of the airliner from the runway during the descent, for OS/ 5 7 . Explain why x(0) = -L and x(7) = 0. 12. Integrate the relation u(f) = deidr and use the condition x(0) = -L to show that x(1)=(1-2")-L. 13. Notice that the time of descent 7 is still unspecified. Use the condition .(7) = 0 to show that 7 = 34In3 Conclude that with _ = 100 mi and to = 450 mi/hr, the time of descent is = 0.37 hours. 14. Knowing x(1) and deidr (as functions of time) along the flight path, it would be nice to know y() and dyair ds well. Proceed as in Step 5, with i - deidr and x given in Steps 10 and 12. Calculate and graph dyit. Where or when along the flight path is the vertical velocity a maximum? A minimum? 15. Carry out Steps 10-14 assuming the horizontal velocity is given by the linear function A(r)= u, + (u, -u. )/T . Does the resulting model give more realistic results than the model with