Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates (-1, -3) and arrived in the Iron Hills at the point with coordinates (1, 0). If he began walking in the direction of the vector V=3i+1j, and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn. ( , )

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be parametric esjuations for a tb) C i v i) C intersects esactly one e ine containing the point -1.3. intersects the xy and ntion voctor to find another esqua ne line with the Korm 17) Let C be the line the ine eontaining the point a) Give parametric ejuastions for c (b) Write an equation for C in on vector to find another equa for the same 18. Let C be the line deterimined by the tt (a) Provide a vector b) Give parametricequations fore c 19. Let C be the line determined by the the points where the line + bt.to t et) eplanes. Ia-0, esplain (a) Provide a vector (b) Write an equation for C in s, determine wbether the the distance between the c. 20. We wish to find the distance from C as shown in the figure that fo dinates of points P and Pa, but we do dinates of point Q (a) Ifyuknew the measure of angle Î¸ llows Wek sect, find the point of intent know th- t.5-3,6+7. weould find the distance from point P to lanes r2()3-,4+3t,5-2 Applications ne. Explain why the vation in Washington State. She tonic wastes in one of the tank tank farm is huge, and she d is leaking. Worse yet, the tank g the tanks would req tank farm, an operation that Instead, Emmy has wells dug the tank farm, to try to trace the earth's surface is conside the bottom of the tank farm the distance is given in f taminated groundwater at (1033, 247,-55). If the w of one of the tanks and i what is the location of th the line does not ow you can distance from point P to line C even if you do ne know the measure of angle Î¸. line containing use the tech- ization for the 24. P023.5), ds(23.5) 25,Pi-1, 3, 7. d = (2,0,4) the leak? eyuivalent.6. Ptbc 53. lan is climbing Mountl He has made a secon mountain. He does n 27, P3, 1), d= (2 5) 28, p(1.3,-2, 4), d= (4,-1,5, 8) In Exercises 29-34, find an equation of the line containing the Proofs ntaning the given pair of points. Express your answer (a) as a vector p ess your an- rametrization, (b) in terms of parametric equations,and ( n 54. Let Ci and C be line symmetric form. 29. P(O,0,0), Q(4,-1,6) and P2 and Qa poin C2 if and only if P â¢P(3,-1,7), Q(5,8,-2)

Math 1113 Fall 2017 . Fulk Project 2 Solve the following problems below individually or in groups of up to three. Each group will submit only one project. Each group member will also submit an explanation their contribution to the group project. Show all work leading to your solutions. To receive credit, you must clearly support you answers. Turn in on Wednesday, November 29 Points may be taken off for sloppy your project at the beginning of class disorganized, or poorly written work 1. (20 points) As viewed from the earth, the angle subtended by the full moon is s 0518Â°. Use this information and the fact that the distance AB from the earth to the moon is 236,900 mi to find the radius of the moon. (20 points) Variable stars are ones whose brightness varies periodically. One of the most visible is R Leonis; its brightness is modeled by the function b) -7.9-2.1 measured in days. a) Find the period of R Leonis. b) Find the maximum and minimum brightness c) Graph the function b. 2. where t is 3. (20 points) A movie theater has a 25-foot-high screen that is 8 feet above eye level. If you sit too close, your viewing angle is too small. If you sit too far back, the image is too small. If you sit x feet back from the screen, your viewing angle, Î¸, is given by Find the viewing angle, in radians, at distances of 5 feet, 10 feet, 15 feet, 20 feet, and 25 feet. Then graph the function in a graphing calculator with a viewing window of [0, 50, 10] by 10, 1, 0.1]. Use the calculator to find the distance at which you have a maximum viewing angle. [Hint: Viewing windows are written as [minmax Xscl by Dmin.Jmax Yscil. These values can be adjusted by pressing the WINDOW button on the calculator. To find a maximum, press 2ND TRACE, choose maximum, move to the left of the maximum and press ENTER, move to the right of the maximum and press ENTER, and then press ENTER again. (x-79j+12, (20 points) The number of hours of daylight in Boston is given by y-3sin where : s he number of days after lanuary 1. Within a year,when does Boston have 13.5 hours of daylight? Give your answer in days after January 1 and round to the nearest day. 4

Page 3 of 5 which gives us y as a function of Theoretically. the integration on the right of t= 1/ - g + alpha f(V)dv can be performed and the result could provide velocity. v,. as a function of t; Then we could perform the integration in y=y0 + V/- g + alpha f(V)dv and substitute v(t) to obtain y as a function of t. Theoretically, that is. Let's look at in example An Example Suppose the air resistance model for a particular object in free fall has where units for v area feet/second Note how f is defined in terms of the sign of v. As usual. downward has been selected as the negative direction so velocities v of an object in free fall will have negative values Thus -v will be positive and the square root makes. sense Similarly if the object is moving upward, v> 0. the value of f(v)) is negative. The object is dropped from a height: of 500 feet with no initial velocity Using g = 32, the integral models from above becomes and where we have used the definition of f(v) for negative values of v since the object is in free fall With a little hard work or the use of a CAS, the first equation can be written as Page 4 of 5 and the second as Now if we could solve the first expression for v in terms of r and substitute the result for v in the expression for y, the result would be very unwieldy to work with. Actually it is impossible to solve the first expression to get a nice closed form for v. Thanks to technology. however, these expressions can still be used to study the object's motion Choose any value of v. say v=-1. Substituting -1 for v in the equations for t and y tells us that t = 0.032 and y = 499.984 so the point (0.032, 499.984) would lie on a graph of distance versus time Similarly, setting v = -50 provides the point (1 839, 452 462) By using a wide range of values for v. we can generate the complete graph of the distance function as a function of time Fortunately, most CAS provides the capability of plotting the points (t(v), y(v)) foe a range of values of v without having to compute each point separately Such a graph appears below. One final note. It is also possible that the integrals representing t and y cannot be integrated into a closed form No problem In this case, simply use the expressions t= 1/ - g + alpha f(V)dV y=y0+ V/ - g + alpha f(V)dV Page 5 of 5 and the numerical integrators provided by a CAS to determine values of t und y Exercises In all of the exercises you will need a CAS or related tool In the example above the graph indicates the object strikes the ground some time between 6 and 7 seconds from release Determine this time to three decimal places In the case where alpha = 1/2 ,f(v) = -v, v0 = 0, y0=100 determine closed form. expressions for v and y as function. Of t. An object is thrown upward at 40 ft/sec from a height of 100 feet The air resistance can be modeled by alpha =0.001 and f(v) = - |v| v Using the trajectory model determine the speed of the object when it strike the ground, determine the time the object strikes the ground after takeoff, construct the graph of the object's height versus time.

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