There are two parts to this assignment. The first part is on WeBWorK — the link is available on the course
webpage. WeBWorK questions can be submitted directly online. They should be completed before you attempt
the second part of the assignment, which consists of the questions on this page. For these questions, you
are expected to provide full solutions with complete arguments and justifications. You will be graded on the
correctness, clarity and elegance of your solutions. Your answers must be typeset or very neatly written.
They must be stapled, with your name and student number at the top of each page.
1. Recall the cone optimization problem done in class: given a circle of paper of radius r, as pictured below,
you cut out a wedge of angle θ and formed the remaining paper into a conical cup whose length from rim
to tip was r.
Now suppose you do the same thing, but form two cups: one from each piece of the original circle.
Determine the angle θ that maximizes the combined volume of the two cups.
2. Suppose two sprinters racing against each other finish in a tie. Explain why the two sprinters must have
both been travelling at the same speed at some particular instant.
3. In this question, you will attempt to describe Dante’s famous phrase Amor, ch’a nullo amato amar
perdona mathematically.
Consider two teenagers in love — call them Dante and Beatrice. Let D(t) be the degree of love felt by
Dante for Beatrice; and B(t), the degree of love felt by Beatrice for Dante. (Note that D and B may be
negative.) Let D and B be modelled by the following system of differential equations:
kD + lB
mD + nB.
(a) Suggest constants k, l, m and n that make the model above as accurate as possible. Justify your
answers briefly.
(b) Given your suggested system of equations, sketch the nullclines on the x-y plane, along with a few
representative trajectories. Be sure to indicate the direction of the trajectories.
(c) What happens, in the long term, to the love between Dante and Beatrice as you have modelled it?