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Trend Lines
• Ex. Suppose the number of students at the University of Arizona since 1990 is given by the following table. Fit several trend lines to the data.
Use each trend line
to predict the number
of students in the years
2004 and 2020.
Years since 1990 Students at UA
0 24,155
2 26,872
4 29,119
6 33,482
8 37,004
10 40,653
Trend Lines
• Linear: Approx. 46,956 students in 2004
Approx. 73,756 students in 2020Linear Trend Line y = 1674.99x + 23505.90
R2 = 0.9910
05,000
10,00015,00020,00025,00030,00035,00040,00045,000
0 2 4 6 8 10
Years Since 1990
Stu
de
nts
Trend Lines
• Quadratic: Approx. 49,976 students in 2004
Approx. 100,478 students in 2020Quadratic Trend Line y = 43.57x2 + 1239.27x + 24086.86
R2 = 0.9968
05,000
10,00015,00020,00025,00030,00035,00040,00045,000
0 2 4 6 8 10
Years Since 1990
Stu
de
nts
Trend Lines
• Exponential: Approx. 50,493 students in 2004
Approx. 117,710 students in 2020
Exponential Trend Liney = 24076.24e0.0529x
R2 = 0.9964
05,000
10,000
15,00020,00025,00030,000
35,00040,00045,000
0 2 4 6 8 10
Years Since 1990
Stu
de
nts
Demand, Revenue, Cost, & Profit
• Ex. Suppose the following data represents the total number of shoes sold in a month at a particular price in
dollars. Use a second
degree polynomial
trend line to find a
formula for the Demand
function
Number of shoes Price
200 $76
350 $68
450 $59
700 $53
900 $40
1100 $24
Demand, Revenue, Cost, & Profit
Demand
D (q ) = -0.0000167q 2 - 0.0326q + 81.47
R2 = 0.9818
0102030405060708090
0 200 400 600 800 1000 1200 1400 1600
quantity
D(q
)
Demand, Revenue, Cost, & Profit
• Generating graph of revenue
•
• Use “Plotting Points” method
• Use interval [0, q] where q is the q-intercept from Demand graph
qDqqR
Demand, Revenue, Cost, & Profit
Revenue
$-$5,000
$10,000$15,000
$20,000$25,000
$30,000$35,000
$40,000
0 200 400 600 800 1000 1200 1400 1600
quantity
Rev
enu
e
Demand, Revenue, Cost, & Profit
• Optimal quantity to maximize revenue is about 800 units.
• Maximum Revenue is about $36,000
• Price should be about $45
Demand, Revenue, Cost, & Profit
• Ex. If the fixed cost is $2000 and the variable cost is $35 per unit, determine a formula for total cost and graph C(q).
• C(q) = 2000 + 35q
Demand, Revenue, Cost, & Profit
Cost
$0
$10,000
$20,000
$30,000
$40,000
$50,000
$60,000
0 200 400 600 800 1000 1200 1400 1600
quantity
Co
st
Demand, Revenue, Cost, & Profit
• Graph of Revenue and Cost (determine profit)
Revenue and Cost
$-
$10,000
$20,000
$30,000
$40,000
$50,000
$60,000
0 200 400 600 800 1000 1200 1400 1600
Quantity
Do
lla
rs
Revenue
Cost
Demand, Revenue, Cost, & Profit
• Profit function: P(q) = R(q) - C(q)Profit
$(20,000)
$(15,000)
$(10,000)
$(5,000)
$-
$5,000
$10,000
$15,000
0 200 400 600 800 1000 1200 1400 1600
quantity
Pro
fit
Demand, Revenue, Cost, & Profit
• Project (Demand)120,000
Market Number Market Size Price
Projected Yearly Sales
(number of drives)
1 1,956,000 $119.95 14,5632 1,044,000 $129.95 7,1433 492,000 $139.95 3,1794 1,512,000 $154.95 8,4045 1,104,000 $169.95 5,2976 1,224,000 $179.95 4,573
Potential national market (in K's):
Test Markets
Demand, Revenue, Cost, & Profit
• Project
- Keep units straight
- Prices (dollars)
- Revenue (millions of dollars)
- Quantities in test markets (whole units)
- Quantities in national market (thousands of units)
Demand, Revenue, Cost, & Profit
• Project (Demand)
- Convert test market data to national data
- Determine quadratic demand trend line (8 decimal places)
populationmarket National
salesmarket National
populationmarket Test
salesmarket Test
Demand, Revenue, Cost, & Profit
• Project (Revenue)
- Units should be millions of dollars
- Typically
- Must adjust for units
qDqqR
Demand, Revenue, Cost, & Profit
• Project (Revenue)
Must convert revenue to millions of dollars
***Use this formula
ds)in thousan is(quantity 1000 qDqqR
1000/
/1,000,0001000
qDqqR
qDqqR
Demand, Revenue, Cost, & Profit
• Project (Revenue)
Revenue
$-
$20
$40
$60
$80
$100
$120
0 200 400 600 800 1000 1200 1400 1600
Quantity (in K's)
Do
llar
s (i
n m
illi
on
s)
Demand, Revenue, Cost, & Profit
• Project (Cost)
- Use COST function from Visual Basic Editor
(will be explained in class)
Demand, Revenue, Cost, & Profit
• Project (Cost)
7 parameters for COST functionquantityfixed costbatch size 1batch size 2marginal cost 1marginal cost 2marginal cost 3
$21.60
Cost per drive$115$100$90Further:
QuantityVariable Costs
Fixed Cost For The Year (in millions):
First 500,000 drives:Next 600,000 drives:
Demand, Revenue, Cost, & Profit
• Project (Revenue and Cost)
- Graph both R(q) and C(q)
- Use “plotting points” method
Demand, Revenue, Cost, & Profit
• Project (Revenue and Cost)
Revenue and Cost
$-
$30
$60
$90
$120
$150
0 200 400 600 800 1000 1200 1400 1600
Quantity (in K's)
Do
llar
s (i
n m
illi
on
s)
Demand, Revenue, Cost, & Profit
• Project (Profit)
Profit
$(40)
$(30)
$(20)
$(10)
$-
$10
$20
0 200 400 600 800 1000 1200
Quantity (in K's)
Do
llar
s (i
n m
illi
on
s)
Demand, Revenue, Cost, & Profit
• Project (Revenue and Cost)
- Determine important information from graphs
Break-even pts at about 300,000 and 800,000
units
(zero profit)
Max profit at about 575,000 units
Negative profit: q < 300K and q > 800K
Revenue and Cost
$-
$30
$60
$90
$120
$150
0 200 400 600 800 1000 1200 1400 1600
Quantity (in K's)
Do
llar
s (i
n m
illi
on
s)
Break-even pts
Largest gap = max profit
Demand, Revenue, Cost, & Profit
• Project (Revenue and Cost)
- Determine important information from graphs
Break-even pts at about 300,000 and 800,000
units
(zero profit)
Max profit at about 575,000 units
Negative profit: q < 300K and q > 800K
Profit
$(40)
$(30)
$(20)
$(10)
$-
$10
$20
0 200 400 600 800 1000 1200
Quantity (in K's)
Do
llar
s (i
n m
illi
on
s)
Break-even pts
Max profit
Demand, Revenue, Cost, & Profit
• Project (What to do)
- Create Demand graph using trend lines
- Create Revenue and Cost graph
- Create Profit graph
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