application linear function
TRANSCRIPT
2007 Pearson Education Asia
Applications and Linear FunctionsApplications and Linear FunctionsExample 1 – Production Levels
Suppose that a manufacturer uses 100 lb of material to produce products A and B, which require 4 lb and 2 lb of material per unit, respectively.
Solution: If x and y denote the number of units produced of A and B, respectively,
Solving for y gives
0, where10024 yxyx
502 xy
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Demand and Supply Curves
• Demand and supply curves have the following trends:
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Demand Function
• Relationship between demand amount of product and other influenced variables as product price, promotion, appetite/taste, quality and other variable.
• Q = f(x1,x2,x3,……xn)
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Demand Function
D : Q = a –b P
Q P20 10018 20016 30014 40012 50010 600
100 200 300 400 500 600
10
121416182022
Q
P
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Linear Demand function
Q = a - b P
Q : amount of product
P : product price
b : slope ( - )
a : value of Q if P = 0 P
Q
0
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Property of Demand function1. Value of q and p always positive or >= 0
2. Function is twosome/two together, each value of Q have one the value of P, and each value of P have one the value of Q.
3. Function moving down from left to the right side monotonously
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Supply function
• Relationship between Supply amount of product and other influenced variables as product price, technology, promotion, quality and other variable.
• Q = f(x1,x2,x3,……xn)
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Supply Function
S : Q = a +b P
Q P10 10012 20014 30016 40018 50020 600
100200 300400 500 600
10
121416182022
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Linear Function Supply
Q = a + b P
Q : Amount of product
P : product orice
b : slope ( + )
a : value of Q if P = 0 P
Q
0
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Property of Supply Function1. Value of q and p always positive or >= 0
2. Function is twosome/two together, each value of Q have one the value of P, and each value of P have one the value of Q.
3. Function moving up from the left to the right side monotonously
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The point of market equilibrium• Agreement between buyer and seller
directly or indrectly to make the transaction of product with certain price and amount of quantity.
• In mathematics the same like crossing between demand and supply function
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Equilibrium
• The point of equilibrium is where demand and supply curves intersect.
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• D: P = - 2 Q + 10
• S :P = 3/2 Q +3
• A. Determine equilibrium point
• B. Graph D, S function
• C. Determine the interval value of P and Q in term of D and S function
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Exercise : Price - DemandAt the beginning of the twenty-first century, the world
demand for crude oil was about 75 million barrels per day and the price of a barrel fluctuated between $20 and $40. Suppose that the daily demand for crude oil is 76.1 million barrels when the price is $25.52 per barrel and this demand drops to 74.9 million barrels when the price rises to $33.68. Assuming a linear relationship between the demand x and the price p, find a linear function in the form p = ax + b that models the price – demand relationship for crude oil. Use this model to predict the demand if the price rises to $39.12 per barrel.
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Exercise : Price - DemandSuppose that the daily supply for crude oil is 73.4 million
barrels when the price is $23.84 per barrel and this supply rises to 77.4 million barrels when the price rises to $34.2. Assuming a linear relationship between the demand x and the price p, find a linear function in the form p = ax + b that models the price – demand relationship for crude oil. Use this model to predict the supply if the price drops to $20.98 per barrel.
What’s equilibrium point and make a graph in the same coordinate axes
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Example 1 – Tax Effect on Equilibrium
Let be the supply equation for a manufacturer’s product, and suppose the demand equation is .
a. If a tax of $1.50 per unit is to be imposed on the manufacturer, how will the original equilibrium price be affected if the demand remains the same?
b. Determine the total revenue obtained by the manufacturer at the equilibrium point both before and after the tax.
50100
8 qp
65100
7 qp
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Solution: a. By substitution,Before tax, and
After new tax, and
100
50100
865100
7
q
qq 5850100100
8p
70.5850.51)90(100
8p
90
65100
750.51100
8
q
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Solution: b.Total revenue given byBefore tax
After tax,
580010058 pqyTR
52839070.58 pqyTR
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BREAK EVENT POINT
• BEP is identifying the level of operation or level output that would result in a zero profit. The other way thatr the firm can’t get profit or don’t have loss
• TC= FC + VC
• TC : Total Cost
• FC : Fixed Cost
• VC : Variabel Cost
• VC = Pp x Q = cost production per unit x
• amount of product
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• TR = Pj x Q
• Tr : Total Revenue
• Pj : Selling Price
• Q : Amount of product
Profit = TR –TC
BEP TR=TC
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BEP
TR
TC
FC
Q
$
0 Q BEPp
C BEP
loss
profit
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Example 2 – Break-Even Point, Profit, and Loss
A manufacturer sells a product at $8 per unit, selling all that is produced. Fixed cost is $5000 and variable cost per unit is 22/9 (dollars).a. Find the total output and revenue at the break-even
point. b. Find the profit when 1800 units are produced.c. Find the loss when 450 units are produced.d. Find the output required to obtain a profit of $10,000.
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Break-Even Points
• Profit (or loss) = total revenue(TR) – total cost(TC)
• Total cost = variable cost + fixed cost
• The break-even point is where TR = TC.
FCVCTC yyy
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Solution:a. We have
At break-even point,
and
b. The profit is $5000.
5000922
8
qyyy
qy
FCVCTC
TR
900
50009228
q
yy TCTR
72009008 TRy
50005000180092218008
TCTR yy
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BEP Exercise• A firm produce some products where the cost per unit is
Rp 4.000,- and selling price per unit is Rp12.000,-.Management developed that fixed cost is Rp 2.000.000,-Determine the amount of product where the firm should sell amount of product so that the break event point achieved.
• a. Find the total output and revenue at the break-even point.
• b. Find the profit when 1600 units are produced.
• c. Find the loss when 350 units are produced.
• d. Find the output required to obtain a profit of Rp 7,000.