Download Semester 1 First Assignment 2016 This assignment comprises three

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