Math 142: Lecture 2-mathematical models
Yanfang Yang
June 3rd, 2015
Definition 0.1. To solve any applied problem, we must first take the problem and translate it into
mathematical problem. This requires us to create equations and functions. This is called Mathematical
Modeling.
Remark 0.1. There are 3 steps in mathematical modeling:
• Formulate the model
State the question.
Identify important factors.
Formulate a mathematical description.
• Manipulate the mathematical formulation.
• Evaluate the model.
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Cost, Revenue and Profit
Any manufacturing firm has two types of costs: fixed and variable.
FIXED COSTS are those that do not depend on the amount of production.
e.g., Real estate taxes, rentals, some management salaries, certain minimal maintenance...
VARIABLE COSTS depend on the amount of production.
e.g., cost of material and labor.
Cost=(variable cost)+ (fixed cost).
In the Linear cost model we assume that the cost m of manufacturing one unit is the same no
mattter how many units are produced. Thus, the variable cost is the number of units produced time the
cost of each unit:
variable cost = cost per unit(m) × number of units produced(x)
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If b is the fixed cost and C(x) is the cost, then we have following:
C(x) =cost
=(variable cost) + (fixed cost)
=mx + b
Notice that we must have C(x) ≥ 0. The y-intercept is the fixed cost and the slope is the cost per item.
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.
If we denote the revenue by R(x)
R(x) =revenue
=(price per unit) × (number sold)
=px
Note 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.
Regardless of whether our models of cost and revenue are linear or not, profit P (x) is always revenue
minus cost Thus
P (x) =profit
=(revenue) − (cost)
=R(x) − C(x)
Example 1.1. (Tomastik) In 1996, Rogers and Akridge of Purdue University studied fertilizer plants in
Indiana. For a typical small-sized plant, they estimated fixed costs at $235,487 and estimated that it costs
$206.68 to produce each ton of fertilizer. The plant sells its fertilizer output at $266.67 per ton. Find the
cost, revenue, and profit equations.
Definition 1.1. The break-even quantity is the value of x at which the profit is zero. That is, it is
the quantity at which revenue and costs are equal.
Example 1.2. Find the break-even quantity for the previous example.
Example 1.3. If a linear cost function is given by C(x) = 10x + 8 and the revenue function is given by
R(x) = 12x, then
• Find the profit function P(x);
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• Find P(1);
• Find P(5);
• Find the break-even quantity.
Example 1.4. The monthly cost of driving a truck depends on the number of miles driven. A truck
owner calculated that in April it cost him $60 to drive 320 miles and in May it cost him $140 to drive 480
miles. Assuming the relation is linear, express the monthly cost C as a linear function of the distance d
driven.
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Supply, Demand and the Equilibrium point
Problem 2.1. 1. If the price of a particular product decrease, would you, as a customer or consumer,
buy more of that product?
2. If the price of a particular product increase and youre the owner of the factory which produces this
product, would you supply more of that product to the market?
Definition 2.1. Assume that x is the number of units produced and sold by the entire industry during a
given time period. Assume also 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.
Remark 2.1. Demand curve is tpically decreasing
Example 2.1. When the price for an item is set at $10, consumers are willing to buy 20 items. When the
price is set at $20, no consumer is willing to buy the item. Find the linear demand equation, p = D(x).
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.
Example 2.2. The producer is not willing to sell any items if the price is set at $10. However, the
producer will supply 20 items when the price is $20. Find the linear supply equation, P = S(x).
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The point at which the demand and supply curves intersect each other is called the equilibrium point.
The x-coordinate is called the equilibrium quantity and the p-coordinate is called the equilibrium
price.
Example 2.3. Demand: p = -x + 15 and supply: p = x + 5 are given. Find the equilibrium quantity
and price.
Example 2.4. It is found that the consumers will demand 60 T-shirts when the unit price is $30 whereas
they will demand 310 T-shirts when the unit price is $5. Assuming that the demand function is linear,
and the selling price is determined by the demand function.
a) Find the demand equation.
b) Find the revenue function.
c) Find the number of items sold that will give the maximum revenue. What is the maximum revenue?
d) If the company has a fixed cost of $200 and a variable cost of $15 per T-shirt, find the companys
linear cost function.
e) What is the companys maximum profit?
f ) How many T-shirts should be sold for the company to break even?
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