Answer Happy

REQUESTING HELP FOR QUESTION #4:"Graph the approach path if the conditions stated in Problem 3 aresatisfied"
Applied Project Where Should a Pilot Start Descent? An approach path for an aircraft landing is shown in the figure and satisfies the following conditions: i. The cruising altitude is h when descent starts at a horizontal distance from touchdown at the origin.. ii. The pilot must maintain a constant horizontal speed throughout descent. iii. The absolute value of the vertical acceleration should not exceed a constant k (which is much less than the acceleration due to gravity). y=Pixl 10 1. Find a cubic polynomial P(x) = ax³ + bx² + cz+d that satisfies condition (i) by imposing suitable conditions on P(z) and P' (a) at the start of descent and at touchdown. 2. Use conditions (ii) and (iii) to show that 6h² -sk 2 3. Suppose that an airline decides not to allow vertical acceleration of a plane to exceed k = 860 mi/h². If the cruising altitude of a plane is 35,000 ft and the speed is 300 mi/h, how far away from the airport should the pilot start descent? 4. Graph the approach path if the conditions stated in Problem 3 are satisfied. James Gregory The first person to formulate the Chain Rule was the Scottish mathematician James Gregory (1638-1675), who also designed the first practical reflecting telescope. Gregory discovered the basic ideas of calculus at about the same time as Newton. He became the first Professor of Mathematics at the University of St. Andrews and later held the same position at the University of Edinburgh. But one year after accepting that position, he died at the age of 36. Chapter 3: Differentiation Rules: Applied Project Where Should a Pilot Start Descent? Book Title: Calculus Early Transcendentals Printed By: Derek Lennen ([email protected]) e2021 Cengage Leaming. Cengage Learning Ⓒ2022 Cengage Leaming Inc. All rights reserved. No part of this work may by reproduced or used in any form or by any means-graphic, electronic, or mechanical, or in any other manner-without the written permission of the copyright holder.

What we want: To Know When a Pilot Should stort his or her descent What we are given: i. The air Plane's attitude is 'h' units. above the ground and has a horizontal distance 'I units from the touchdown Point to the ground directing under the AirPione 11. The airplane's speet of 'V' must remain.. constant throughout the descent ii. The absorte value of the vertical acceleration Cannot exceed 'K'. Because we were given speed as 'V' and we know twit the derrivative of Seeed 15 acceleration we con model vertiol Occeleration as V₂₂ Se Vs K (0,0) OVE LOV jo Vy ib y-Pon 0 = 30L +26 -26=3aL 3L There are 4 auestions we must answer using the Information above P(0)= a(0)³ + b(0)* + C(0) + d 6/6 (PLO)=d), (d=0) Plato Px)= 3ax² + 26x +( P'(o)= 3a(0) + 2b (0) + C P'rol= C) CEO b/c P'(o)=0 NOW for the L's P(L) = 3a(L²² +26(4) Q=L(3aL+26) 3x Lasting, PCL)= a(L)³ + 6 (L)² h = 92²³² +6²² h= = -264² +6L² us contiued on next Page difcentiated vid / power cure projet, nue joesn't EPPY secause a,b,c, and Represent constient's b/C (=Q & d=0 we can write a 'cleaner" function P(x) = ax² + bx² We won't want that b term. 5/2 I KN/W P'(L)=0 /m going to set érmation ever to o and Sorve for a col Love 11453 UNKNOWN WHICH WIR LEIP WHEN SOLVING PL 6/C P(L) = h now Pinternt in a and Multiping w/ the i lets cances an L from num. I ten, 1. we have to find a that satisfies condition i. suitable conditions on PCXI tescent starts and during P(x) = ax³+ 6x² +CX +d P²(0)=0 P(0)=0 when what tus grope iirustrates are the points the descent begins at P(L)=h the slope of the itne tanget to the function y=P(x) at PEL)=h which is MED. DC P'(L)=0 as well as the point when the plane lands or PCO)=0 45 won as it's tangent which is also 0 of ²010 P(0)=0 P'(0)=0 P(L) = h h == 26L² + 6L² =-3h+b P'(L)=0 I am going to start by pingging PLO)=0 then will into P(x) = ax³ + bx²+x+d feres de 2/7=6( - 3+1) -26 34 Q=- er a=2₁34 L P y=PLA) we have our Pollo romial Muiting both sides by 3 (6 = 24 The Now we can ping binto a inside of a the function and plug in p²(0)=0 Cubic Poinomial by imposing When and plexi touch down. th -2h 3L a P'(L)=0 PELI=h in divide on L factor ont b P(x) = - and out b unknown is now known 35 concel to get rid of the b So we can model the P(x) = ax³ + bx² - 2 + x³ + 3+ x²

2. Now we need to show that the sk using conditions i. & i. A condition it. Stoves that horizontal speed must be consient since the pille is moving toward the left or negotive site speed which was given as 'V² 13. actually V of the change in x with respect to rime which word be = -√ Condtion states that the abisonte vaine of verrical acceleration can't exceed k we must establish: P(X)=y PCX1=-x²+x² (1=1 and -V # ['가 끝x?] Continuel on next page JVýsk Firsting we need to So S 3. We can't ret vertical acceleration exceed k= 860 M/L² /6 cruising altitude is 25,000 Feet and cruisme speed is 300 milh How For from the airport showed the pilot Start descenting L = 6(6.628) (300) 960 L = 64,5 M rate of veraces chante convert 35000 feet into miles 35,000 5230 = 6.628m²=h V=300mi/ k = 860/1² poving this into instag the Revious formula for K we con soive for L which will tell us hoir for from the alreact we should be considering L represents distance. K₂ GAy ² T multiling buth sides by I' and divile 55 K Square root 12= 6x² Ch x dt² - 12hVxx hv. dx L at H 1²-12 hv²x 6hv² Jt² L' *(4) = -12h/²+1/² av=-Chy² ke hy² 10,1 = 6/² layl≤k) ✓ of This is the bernivorite Dosition which veineity differentide get again to get the vertical acceleration just when I thought I was getting Liesniz nord.com... I'm going to bemore at (0,01 bic I have 2's and at L we know aceleration is of a MINIMUM This stored in the rest negative vertico acceleration 나. GMP the approach path Remember that PCx) = -24x²+ 3+x²