2.3 Functions and Mathematical
Models
By
Dr. Julia Arnold
A mathematical model is a mathematical representation of a
real-world problem.
Some problems may approximate a linear function:
y = mx + b
Others may be better approximated by some polynomial function
or power function such as our Cost function in lesson 2.2
While still others may require a rational function such as the
model for driving costs based on 1992 model compact cars which
was found to be C(x) = 2095  20 . 08
x
2 .2
Economic Models
A demand equation expresses the relationship between the unit
price and the quantity demanded. The graph of the demand
equation is called a demand curve.
For example: At Christmas time the demand for certain popular items
sometimes exceeds the manufacturer’s ability to provide them. Remember the
wii?
The equation that expresses the relation between the unit price
and the quantity supplied is called a supply
equation and its graph is called a supply curve.
If the wii starts saturating the market, maybe it
will cost below the $199 sales tag it has right now. This will mean that the
supply is sitting on the shelf and is not
being demanded by the consumer.
Market equilibrium prevails when the quantity produced is equal
to the quantity demanded and the corresponding price is called
the equilibrium price. If the wii keeps selling out as the store keeps
restocking then there is an equilibrium between supply and demand.
For the demand equations where x represents the quantity
demanded in units of a thousand and p is the unit price in dollars,
a) sketch the demand curve and b) determine the quantity
demanded when the unit price is set at $p.
2
p   x  36 ; p  11
y
This side
represents
negative x
values
which
would be
invalid.
Only this
number
would be
our
solution.
2
11   x  36
2
x  25
x 5
5,000 since
X is in units of
A thousand.
Blue line is p = 11
x
Assume that the demand function for a certain commodity
has the form
p
2
 ax  b , a  0 , b  0
where x is the quantity demanded, measured in units of a
thousand and p is the unit price in dollars. Suppose the
quantity demanded is 6000 (x = 6) when the unit price is $8
and 8000 (x = 8) when the unit price is $6. Determine the
demand equation. What is the quantity demanded when the
unit price is set at $7.50?
Assume that the demand function for a certain commodity has
the form
2
p   ax  b , a  0 , b  0
where x is the quantity demanded, measured in units of a
thousand and p is the unit price in dollars. Suppose the
quantity demanded is 6000 (x = 6) when the unit price is $8
and 8000 (x = 8) when the unit price is $6. Determine the
demand equation. What is the quantity demanded when the
unit price is set at $7.50?
2
8
 36 a  b
64   36 a  b
64  36 a  b
2
6
 64 a  b
36   64 a  b
36  64 a  b
8
 a (6 )  b
6
 a (8 )  b
64  36 a  36  64 a
28  28 a
1a
64 + 36(1) = b
100 = b
p
2
 x  100
7 .5 
2
 x  100
2
56 . 25   x  100
2
x  43 . 75
x  6 . 614
6 , 614 units
For each pair of supply and demand equations where x
represents the quantity demanded in units of a thousand and p
the unit price in dollars, find the equilibrium quantity and the
equilibrium price.
2
p   x  2 x  100
p  8 x  25
For each pair of supply and demand equations where x
represents the quantity demanded in units of a thousand and p
the unit price in dollars, find the equilibrium quantity and the
equilibrium price.
2
p   x  2 x  100
p  8 x  25
2
8 x  25   x  2 x  100
2
x  10 x  75  0
2
x  10 x  75  0
x  15 x  5   0
x   15 , x  5
Since -15 would not make sense, the equilibrium quantity is 5000
and the equilibrium price is 8(5)+25=$65