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(Solved) Introduction to Financial Mathematics - Semester 1, 2017 Assignment 9 Algebra To be handed in by 2pm, Monday 22nd of May Examples: Graph the solution...


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Introduction to Financial Mathematics – Semester 1, 2017
Assignment 9 Algebra
To be handed in by 2pm, Monday 22nd of May
Examples:
A. Graph the solution to the system of inequalities
x + 5y ≤ 20, 4x + 5y < 60 with x ≥ 0 and y ≥ 0. State whether the region is convex, closed and/or bounded.
B. Formulate as a linear program, but do not solve.
Three stocks have been analysed in terms of growth in market value during
the next year (short-term growth), growth in market value over the next 12
years (long-term growth), and anticipated dividend rate. The data appear in
the table below, in terms of the percentage of the amount initially invested.
Stock 1
Anticipated growth rate during next year
3.2%
Anticipated growth rate in next 12 years
100%
Anticipated (annual) dividend rate
3.0% Stock 2 Stock 3
3%
2.5%
140%
70%
3.6%
3.9% The investor wishes to determine the minimum amount that should be invested while satisfying the following conditions:
(i) appreciation of at least $2000 over the next year (this is not re-invested);
(ii) appreciation of at least $30000 over the next 12 years (this is not reinvested);
(iii) dividends of at least $1500 per year.
Write down the inequalities that describe the constraints and the linear
function to be minimised.
Questions:
1. Suppose that a bounded closed convex region has vertices (0, 0), (0, 2),
(1, 0), (10, 6). Find the maximum and minimum value of f (x, y) = 3x − 5y
in this region.
2. Graph the solution of the following systems of inequalities, and for each
state whether it is convex, whether it is closed and whether it is bounded.
(a) y ≤ 2x + 2, y ≤ −x + 3 with x ≥ 0 and y ≥ 0.
(b) y ≥ x, y < −2x + 4 with x ≥ 0 and y ≥ 0.
(c) y ≤ 2x + 3, y ≥ −4x + 6 with x ≥ 0 and y ≥ 0.
(d) y < 2x + 1, 3y > x − 2 with x ≥ 0 and y ≥ 0.
3. Rewrite the following in standard form, where possible. If it cannot be
rewritten in standard form, then explain why not. (a) Maximise x + 2y with respect to the restrictions in 2(a).
(b) Minimise 2x − 3y with respect to the restrictions in 2(b).
(c) Minimise x + 2y with respect to the restrictions in 2(c).
(d) Maximise −3x + y with respect to the restrictions in 2(d).
4. Find the maximum and minimum of the linear function f (x, y) = x − y/2 in
the region of Question 2(c), if possible. If they exist, where do they occur?
If they do not exist, why not?
5. A pet food manufacturer produces two types of food, regular and premium.
A 10 kg bag of regular food requires 3 hours to prepare and 2 hours to cook;
a 10 kg bag of premium food requires 5 hours to prepare and 2 hours to
cook. The materials used to prepare the food are available 12 hours per day
and the oven used to cook the food is available 6 hours per day. The profit
on a 10 kg bag of regular food is $25 and on a 10 kg bag of premium food it
is $30.
(a) Write the inequalities that describe the constraints.
(b) Graph the solution to the system of inequalities and find the vertices of
the region.
(c) Find how many bags of each type of food should be made to maximise
the profit. (Fractional bags can be produced, they just save the leftovers
for the next day.)

 


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