ASSIGNMENT 10
There are two parts to this assignment. The first part is on WeBWorK — the link is available on the course
webpage. The second part consists of the questions on this page. You are expected to provide full solutions
with complete justifications. You will be graded on the mathematical, logical and grammatical coherence and
elegance of your solutions. Your solutions must be typed, with your name and student number at the top of
the first page. If your solutions are on multiple pages, the pages must be stapled together.
Your written assignment must be handed in before the start of your recitation on Friday, November
21. The online assignment will close at 9:00 a.m. on Friday, November 21.
1. At what value(s) of x on the curve y = −3x5 + 160x3 − 2 does the tangent line have the largest slope?
2. (a) Prove that the triangle of maximal area which can be inscribed in a circle of radius 1 is equilateral.
(Hint: first prove that the triangle must be isosceles.)
(b) An n-gon is a polygon with n sides. An n-gon is regular if the sides and angles are all equal. Prove
that if an n-gon inscribed in a circle of radius 1 is not regular, then there is an n-gon of larger area
which can also be inscribed in the circle.
(c) Can you conclude from part (b) that the n-gon of maximal area which can be inscribed in a circle of
radius 1 is regular? Explain your answer in two or three sentences.
3. This question refers to the 2014 paper “The expanding fireball of Nova Delphini 2013” [1]. A copy of the
paper is available on the course webpage.
In the paper, the authors use a model which describes Nova Delphini 2013, a supernova 4.54 kiloparsecs
away, in two components: a spherical core, or “optically thick pseudophotosphere”; and a spherical
outer shell, or “optically thin halo”. The angular diameter of the core is described in equation (1) in
the “Methods” section. In the same section, the authors cite an angular expansion rate dθ
dt of 0.156
milliarcseconds per day.
Find the rate, in cubic kiloparsecs per day, at which the volume of the spherical core is expanding when
the diameter of the core (the actual diameter, not the angular diameter) is equal to d0 . (Your answer
should be an expression involving the constant d0 .)
4. [Bonus] Draw a picture that describes how you feel about mathematics.
[1] G.H. Schaefer et al. “The expanding fireball of Nova Delphini 2013.” Nature (2014), Web. 26 October 2014.