Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
THE UNIVERSITY OF SYDNEY MATH1111 INTRODUCTION TO CALCULUS Semester 1 First Assignment 2016 This assignment comprises three questions and is worth 5% of the overall assessment. It should be completed, scanned and uploaded using Turnitin through the MATH1111 Blackboard portal by 11 am on Monday 11 April. Your answers should be concise, well spaced, legible and a pleasure to read. Please show all working, and present your arguments clearly using words of explanation and diagrams where relevant. Mathematics involves communication of ideas; this is a skill that takes time and effort to master. The first question involves elementary properties of triangles and angles, and is intended to be straightforward, but nevertheless requires clear thinking. The second question involves a real-world application of the Theorem of Pythagoras, manipulations of algebraic expressions and evaluation of formulae. The third question is challenging, and requires a thorough understanding of the geometry of the circle and its relationship with elementary trigonometry, with a rather surprising application to the geometry of the earth and lines of sight. Your tutor will give you feedback and allocate an overall letter grade (and mark) using the following criteria: A+ (10): excellent and scholarly work, answering all parts of all questions, with clear and accurate explanations and working, with appropriate acknowledgement of sources, if appropriate, and at most minor or trivial errors or omissions; A(9): excellent work, making progress on all questions, but with one or two substantial omissions, errors or misunderstandings overall; B+ (8): very good work, making progress on all questions, but with three or four substantial omissions, errors or misunderstandings overall; B(7): good work, making substantial progress on at least two questions, but making five or six substantial omissions, errors or misunderstandings overall; C(6): reasonable attempt, making substantial progress on at least two questions, but making more than six substantial omissions, errors or misunderstandings overall; D(4): making substantial progress on just one question; E(2): some attempt, but making no substantial progress on any question; F(0): no real attempt at any question. 1. In this exercise we explore consequences of an idea involving a line touching a triangle, to explain why the angles of any triangle add up to 180◦ . Consider the following diagram where there is a straight line passing through P and Q, touching triangle ABC at vertex B. Q b B b P θ2 θ1 b β α b A γ b C (a) Write down the angle sum θ1 + β + θ2 in degrees. (b) Redraw the diagram so that the line through P and Q continues to touch the triangle at B but is now parallel to the side AC. Write down any relationships that you notice between the angles α, γ, θ1 and θ2 in your new diagram. (c) Deduce from (a) and (b) that the internal angles of a triangle add up to 180◦ . (d) Comment about whether your diagram in (b) is sufficiently general to justify the conclusion in (c) for all triangles. Draw further diagrams to check cases when the internal angles of the triangle are not all acute. 2. A vertical flag pole of flat roof of a building. metres. The flag pole top to the flag pole at height h metres is erected exactly in the middle of the The roof is rectangular of width w metres and depth d is stabilised by cables that join the corners of the roof a point k metres below the top of the flagpole. b km hm dm wm (a) Let ℓ m be the total length of cable required to stabilise the flag pole. Use right-angled triangles (and the Theorem of Pythagoras) to explain why ℓ = 4 r w 2 + d2 + (h − k)2 . 4 (b) Give a simplified exact surd expression for ℓ given that w = 12, d = 10, h = 8 and k = 3. Now estimate, to the nearest metre, the total length of cable required to stabilise the flag pole, under these assumptions. (c) Rearrange the formula in (a) to express h in terms of ℓ, w, d and k. (d) Estimate, to the nearest metre, the height of the tallest flagpole that can be stabilised using a total of 50 metres of cable, given that the roof top is 12 metres wide, 10 metres in depth, and the point where the cables are fastened to the flagpole is 3 metres from the top of the pole. 3. The Eiffel Tower is 394 metres high. An observer is standing upright, looking directly at the Eiffel Tower, and has eye level 157 cm from the ground. Imagine, for the purposes of this exercise, that the earth is perfectly spherical. The radius of the earth is R = 6, 370 kilometres (to 3 significant figures). Suppose the observer has moved away so that the Eiffel Tower is almost completely obscured by the curvature of the earth, so that looking towards the Tower the observer can see only its very tip. (If the observer moves any further away then the Tower disappears completely from sight.) (a) Find, to three significant figures, the distance (as a straight line) from the observer’s eyes to the tip of the Eiffel Tower. (b) Find, to three significant figures, the curved distance directly along the surface of the earth from the observer’s feet to the base of the Eiffel Tower. (This will be the length of a certain arc of a circle of radius R, as suggested by the the diagram below.) You should include enough information that the reader can see clearly how you derived your answers. (The diagram below is certainly not to scale.) 394 m R km 157 cm R km b centre of the earth