Trend Lines Ex. Suppose the number of students at the University of Arizona since 1990 is given by...

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

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

)


• Generating graph of revenue

•

• Use “Plotting Points” method

• Use interval [0, q] where q is the q-intercept from Demand graph

qDqqR


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


• Optimal quantity to maximize revenue is about 800 units.

• Maximum Revenue is about $36,000

• Price should be about $45


• 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


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


• 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


• 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


• 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


• Project

- Keep units straight

- Prices (dollars)

- Revenue (millions of dollars)

- Quantities in test markets (whole units)

- Quantities in national market (thousands of units)


• 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


• Project (Revenue)

- Units should be millions of dollars

- Typically

- Must adjust for units

qDqqR



Must convert revenue to millions of dollars

***Use this formula

ds)in thousan is(quantity 1000 qDqqR

1000/

/1,000,0001000

qDqqR

qDqqR



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)


• Project (Cost)

- Use COST function from Visual Basic Editor

(will be explained in class)


• 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:


• Project (Revenue and Cost)

- Graph both R(q) and C(q)

- Use “plotting points” method



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)


• 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)



- 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



- 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


• Project (What to do)

- Create Demand graph using trend lines

- Create Revenue and Cost graph

- Create Profit graph

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Trend Lines Ex. Suppose the number of students at the University of Arizona since 1990 is given by...

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