Math 142 Business Mathematics II
Minh Kha
Section 1.2: Mathematical Modeling
Goals: to understand what a mathematical model is, and some of its examples in business.
Definition 0.1. Mathematical Modeling is an attempt to describe some part of the real world in
mathematical terms. More precisely, to solve any applied problem by a mathematical model, we must
first take the problem and translate it into mathematics by creating equations and functions.
There are three steps in mathematical modeling:
1. Formulate the model
• State the question.
• Identify important factors.
• Formulate a mathematical description.
2. Manipulate the mathematical formulation.
3. Evaluate the model.
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Mathematical Models of Cost, Revenue and Profit
There are two types of costs in any manufacturing firm: fixed costs and variable costs.
Fixed Costs are those that do not depend on the amount of production.
e.g., taxes, interest, some salaries, maintenance, protection
Variable Costs depend on the amount of production.
e.g., cost of material and labor
Cost is the sum of fixed and variable cost, i.e., Cost=Fixed cost + Variable Cost.
In the linear cost model, we assume that the cost m of producing one unit is the same no matter
how many units are produced. Thus the variable cost is the number of units produced times the cost
of each unit:
Variable Cost = Cost per unit(m) × Number of units produced(x)=mx.
If b is the fixed cost and C(x) is the cost, then its formula and graph are:
C(x) = Variable Cost + Fixed Cost = mx + b.
(1)
Notice that we must have C(x) ≥ 0. The y-intercept is the fixed cost and the slope is the cost per
item.
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In the linear revenue model, we assume that the price p of a unit sold by a firm is the same no
matter how many units are sold. Thus, the revenue is the price per unit times the number of units
sold. Let x be the number of units sold. (For convenience, we assume the number of units sold equals
the number of units produced.) Then, if we denote the revenue by R(x), then its formula is
R(x)=revenue=(price per unit) × (number sold)=px.
Notice that we must have R(x) ≥ 0. The line goes through (0, 0) because nothing sold results in no
revenue. The slope is the price per unit.
Thus, the profit P is always revenue less cost, i.e.,
P =profit=(revenue)-(cost)=R − C.
Recall that both C(x) and R(x) must be nonnegative functions. However, profit P (x) can be positive
or negative. Negative profits are called losses.
Example 1. (J.D.Kim) A manufacturer of garbage disposals, has a monthly fixed cost of 15, 000 dollars
and a production cost of 30 dollars for each garbage disposal manufactured. The unit sell for 60 dollars
each.
1. What is the cost function?
2. What is the revenue function?
3. What is the profit function?
Definition 1.1. The value of x at which the profit is zero is called the break-even quantity. The
break-even quantity is the value of x at which cost equals revenue.
Example 2. Find the break-even quantity for the previous example.
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Mathematical Models of Supply and Demand
Definition 2.1. Assume that x is the number of units produced and sold by the entire industry during
a given period of time and that p = −cx + d, c > 0, is the price of one unit if x units are sold; that is,
p = −cx+d is the price of the xth unit sold. We call p = −cx+d the demand equation and the graph
the demand curve. Estimating the demand curve is a fundamental problem for the management of
any comany or business.
Note that the demand curve is typically decreasing.
Example 3. When the price for an item is set at $10, consumers are willing to buy 30 items. When the
price is set at $20, no consumer is willing to buy the item. Find the linear demand equation, p = D(x).
Definition 2.2. The supply equation p = p(x) gives the price p necessary for suppliers to make
available x units to the market. The graph of this equation is called the supply curve.
A reasonable supply curve increases because the suppliers of any product naturally want to sell more
if the price is higher.
Example 4. The producer is not willing to sell any items if the price is set at $5. However, the producer
will supply 10 items when the price is $10. Find the linear supply equation, p = S(x).
Definition 2.3. The best-known law of economics is the law of supply and demand. The point where
the supply curve intersects the demand curve is called the equilibrium point. The x-coordinate of the
equilibrium point is called the equilibrium quantity and the p-coordinate is called the equilibrium
price.
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Example 5. Suppose the price of an item is given by p = − x + 30 when x items are demanded.
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Further, manufacturers are willing to sell x items at the price p = x + 9
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a) What is the surplus or shortage if the price is set at $20?
b) What is the equilibrium quantity? What is the equilibrium price?
Example 6. (Tomastic, #27) Many assets, such as machines or buildings, have a finite useful life
and further more depreciate in value from year to year. For purposes of determining profits and taxes,
various methods of depreciation can be used. In straight-line depreciation we assume that the value
V of the asset is given by a linear equation in time t, say, V = mt + b. The slope m must be negative,
since the value of the asset decreases over time. Consider a new machine that costs $10,000 and has a
useful life of nine years and a scrap value of $1000. Using straight-line depreciation, find the equation
for the value V in terms of t, where t is in years. Find the value after one year and after five years.
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Quadratic Mathematical Models
Definition 3.1. A quadratic is a polynomial function of degree 2: q(x) = ax2 + bx + c, a 6= 0. Every
quadratic function can be placed in standard form a(x − h)2 + k. The point (h, k) is called the vertex
and can be found by using the formulas
h = −b/2a,
k = c − b2 /4a.
We will now consider quadratic mathematical models of revenue and profit. If the number of units of
a product sold by an industry is to increase significantly, then the price per unit will most likely need
to decrease (by supply and demand). Therefore, if the demand equation for an entire industry is the
linear demand equation p = −cx + d with c > 0, then the revenue is:
R = px = (−cx + d)x = −cx2 + dx.
This is a quadratic model.
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Example 7. Suppose that price of an item is given by p = − x + 10 when x items are demanded.
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a) Find the revenue function.
b) When is the revenue maximized?
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