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MTH 108 Class 20 Notes
9.5- Business and Economics Applications
Definition:
Suppose a business manufactures a single product.
1. The total cost of production can be thought of as a function C, where C  x  is
the cost of producing x units.
2. The revenue or money the business takes in from the sale of x units can be
thought of as a function R( x) .
3. The total profit is the money taken in less the money spent. Mathematically,
this can be represented as: P  x   R  x   C  x  .
Example:
Suppose a business makes chairs. For one year the business pays $90,000 in fixed
costs to run its operation. In addition it costs the business $150 to produce each
chair, and each chair sells for $400.
(a) Find the total cost, C  x  , of producing x chairs.
This is C ( x)  90000  150 x .
(b) Find the total revenue, R( x) , from the sale of x chairs.
This is R  x   400 x .
(c) Find the total profit from the production and sale of x chairs.
This is P  x   400 x   90000  150 x   250 x  90000 .
(d) Determine the profit that the business will realize from the production and sale
of 400 chairs.
This is given by P  400  250  400   90000  10000 . Thus, the company will
earn a profit of $10000 from the production and sale of 400 chairs.
(e) How many chairs must the company produce and sell to break even?
For this question we are looking for an x such that P  x   0 . So, we solve
250x  90000  0  250 x  90000  x  360 . Thus, the company must make
360 chairs to break even.
A particular factory manufactures car engines. The factory’s profit from
manufacturing and selling x car engines in a given week is given by:
P( x)  100 x2  13000 x  300000 .
(a) Determine the profit that the factory will realize in a week if it manufactures and sells 110
car engines.
2
Solution: The profit will be P 100  100 110  13000(110)  300000  80000 . So, the
factory would lose $80000 if it manufactured and sold 110 car engines in a week.
Graded Example:
(b) How many car engines must the factory manufacture and sell per week to break even?
Solution: We must solve:
P( x)  0  100 x 2  13000 x  300000  0  100  x 2  130 x  3000   0
100  x  30  x  100   0  100  0 or x  30  0 or x  100  0
 x  30,100.
So, the factory can manufacture and sell 30 car engines and break even, or the factory can
manufacture and sell 100 car engines and break even.
Definition:
As the price of a product varies, so too does the amount of the product that will
sell. We define demand of a product as a function D( p) . The function takes the
price of the product as an input and spits out how much of the product will be
demanded at that price.
Definition:
As the price of a product varies, so too does the amount of the product made
available. We define supply of a product as a function S ( p) . The function takes
price of the product as input and spits out how much of the product will be
available from sellers at that price.
Remark:
Usually, we think of demand as decreasing as price goes up, and we think of
supply as increasing as price goes up. The point of intersection between a supply
and demand function is called the equilibrium point. At this point the amount that
the seller will supply is the same as the amount the consumer will buy.
Example:
The demand for an Xbox 360 (in millions) given its price in dollars is
1
D( p)   p  95 . The supply for Xbox 360’s (in millions) given its price in
10
7
dollars is: S ( p) 
p 5.
30
(a) What will be the demand for Xbox 360’s if the price is $120?
This is given by D(120)  83 . Thus, 83 million Xbox 360’s will be demanded
at a price of $120.
(b) What will be the supply of Xbox 360’s if the price is $120?
This is given by S (120)  23 . Thus, 23 million Xbox 360’s will be supplied
at a price of $120.
(c) Find the equilibrium point of the demand and supply functions for the Xbox
360.
1
7
p  95 and S 
p  5 (Note: We are omitting the
10
30
function notation.) For the equilibrium point we want to know when D  S .
So, we must solve the system:
1

 D   10 p  95

7

p 5
S 
30

D  S


which is easy to solve with substitution. So, we solve:
1

 D   10 p  95 
1

D   p  95

7
1
7


10
p 5  
  p  95 
p5
S 
30
10
30

D  7 p  5
D  S
30



 1

 7

 30   p  95   30  p  5   3 p  2850  7 p  150  10 p  3000  p  300.
 10

 30

We know that D  
1
(300)  95  65 which immediately
10
implies that S  65 . So, the equilibrium point is (300,65) (since (300,65)
will be on the graph of both D(p) and S(p)). This means that the equilibrium
price for an Xbox 360 is $300 and the equilibrium quantity is 65 million units.
We note that when p  300 , D  
Note: If time permits the instructor will also illustrate this problem
graphically.
Note: The system of three equations was actually unnecessary in the above
problem. In particular, one could reason that since supply must equal
1
7
demand for the equilibrium point we must have  p  95 
p  5 . In fact
10
30
this is the approach the book takes on page 680 to a similar problem.
This application section marks the end of Chapter 9 (we don’t go through 9.3 and 9.4 in MTH
108). So, to end this class we will begin chapter 10.
10.1- Radical Expressions, Functions, and Models
We begin chapter 10 with a basic, though important question regarding square roots.
Question to the Class:
How many real number square roots does 4 have?
Answer: The number 4 actually has two real number square roots. In particular, both 2 and -2
are square roots of 4, since when they are multiplied by themselves we get 4.
Remark:
The answer to the above question may be somewhat surprising because one is so
used to hearing that “the square root of 4 equals 2.” The next definition will clear
some of this confusion.
Definition:
The principle square root of a nonnegative number is its nonnegative square root.
The symbol “
” is called a radical sign and is used to indicate the principle
square root of the number over which it appears.
Note: Oftentimes, when one sees 4 , he or she says “the square root of 4”. In actuality this is
a slight abuse of language because 4 means the “principle square root of 4”. This means that
4  2 (since 2 is the only nonnegative square root of 4).
Example:
Find
25
The number 25 actually has two square roots: 5 and -5. However, in this problem
we are asked for the principle square root. So, we have that 25  5 .
Graded Example:
(a) Find 169 .
Solution: The solution is 169  13 .
(b) Find
625
.
196
Solution: The solution is
625 25

.
196 14
(c) How many real square roots does the number -1 have?
Solution: If we multiply any real number by itself, we get a nonnegative real number. So, -1 has
0 real square roots.
Recommended Homework: 9.5- 9-27 odd
Read 10.1