DSC3707-assign01-S1-2023
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School
University of South Africa *
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Course
3707
Subject
Mathematics
Date
Nov 24, 2024
Type
Pages
3
Uploaded by ChancellorFang8501
DSC3707/Ass01/S1/2023
202
3
Assignment 01
Mathematical
Modelling
DSC3707
Semester
1
Department
of
Decision
Sciences
Important Information:
This is an online module.
All study material will be posted on
my
Unisa.
Please activate your
my
Life email address and ensure you have regular access to the
my
Unisa module site DSC3707-23-S1.
This document contains assignment 01.
Bar code
ASSIGNMENT
01
DSC3707
DUE
DATE:
6
March
We
encourage
the
use
of
a
computer
package
to
check
your
answers.
Hints
for
the
use
of
wxMaxima
are
included,
however
you
may
use
any
other
relevant
package
or
procedure.
You
may
even
Copy
and
Paste
the
solutions
you
obtain
from
wxMaxima
directly
into
your
assignment.
Assignment
01
covers
the
first
7
chapters
in
the
textbook
(study
units
1
to
3).
This
assignment
contributes
3
5%
towards
your
semester
mark
and
7
%
towards
the
final
exam
mark.
Question
1
(a)
Suppose
that
the
market
for
a
commodity
is
governed
by
supply
and
demand
sets
defined
as
follows.
The supply set
S
is the set of pairs (
q, p
) for which
2
p
-
9
q
= 30
and the demand set
D
is the set of pairs (
q, p
) for which
3
p
+ 7
q
= 168
,
where the price
p
is in rands per unit quantity
q
. Sketch
S
and
D
and determine the equilibrium set
E
=
S
∩
D
, the supply and demand functions
q
S
, q
D
, and the inverse supply and demand functions
p
S
,
p
D
.
(Answer without wxMaxima, but you can check your answer with wMaxima:
Choose [Equations],
then [Solve algebraic system].
(b) Suppose that the government decides to impose on excise tax of
T
rands on each unit of the commodity
in (a). What price will the consumers end up paying for each unit of the commodity?
(c) Find a formula for the amount of money the government obtains from taxing the commodity in the
manner described in (b). Determine this quantity explicitly when
T
= 2
,
00.
Question 2
Suppose that the supply and demand sets for a particular market are
S
=
{
(
q,p
)
|
p
-
6
q
= 357
}
,
D
=
{
(
q,p
)
|
p
+
q
2
+ 4
q
= 621
}
,
where
q
is the quantity of units sold, and
p
is the price in rands per unit.
(a) Sketch
S
and
D
and determine the equilibrium set
E
=
S
∩
D
.
(wxMaxima: Choose [
Plot2D]
. Choose
[Equations]
, then
[Solve algebraic system]
.)
(b) Suppose the government imposes an excise tax of R33 on each unit sold. What is the new equilibrium
set? How much of the tax is paid by the consumer, and how much by the supplier?
2
DSC3707/Ass/S
1
Question 3
Suppose you invest R60 000 in a special savings account where, for the first eight years, compound interest
of 6% is paid annually at the end of each year and, thereafter, interest is continuously compounded at an
annual equivalent rate of 4%. How much money do you have in the account after 18 years if you remove no
money from it during that period?
Question 4
Suppose that the demand function for a good is
q
D
(
p
) =
11000
2
p
2
+ 3
,
where
q
is the quantity and
p
is the price in rands. If the price is decreased from R10 to R9,50, determine,
using a calculus method, the approximate increase in the quantity sold.
Compare your answer with the
answer obtained by using the obvious method of simple substitution. (Answer without using wxMaxima,
but you can then check your answer using wxMaxima.)
Question 5
A population, currently 10000, is growing at 5% per year.
(a) Write a formula for the population,
P
, as a function of time,
t
years in the future.
(b) Estimate the population 10 years from now.
(c) Estimate the doubling time of the population.
Question 6
Suppose that the supply and demand sets for a certain good are
S
=
{
(
q,p
)
|
3
p
-
q
= 30
}
,
D
=
{
(
q,p
)
|
4
p
+
q
= 96
}
,
and suppliers operate according to the cobweb model. That is, if
p
t
and
q
t
are (respectively) the price and
quantity in year
t
, then
p
t
=
p
D
(
q
t
) and
q
t
=
q
S
(
p
t
−
1
). Suppose also that the initial price is
p
0
= 12.
(a) Derive a recurrence equation for
p
t
, and solve the recurrence equation.
(b) How does
p
t
behave as
t
tends to infinity?
(c) How does
q
t
behave as
t
tends to infinity?
(d) Use your results to determine
S
∩
D
, and then check the correctness of your answers for (b) and (c).
3
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