topic 1 slope and rate of change unit 8 topic 1. explore susan is shopping for the best buy on paper...

Topic 1Slope and Rate of Change

Unit 8 Topic 1

ExploreSusan is shopping for the best buy on paper towels. The same brand of paper towels comes in the four sizes shown below.

1. Which is the better deal?

2. What methods did you use to determine which is the better deal?

3. Why would someone want to buy the more expensive option?

Explore

1. Which is the better deal?

The better deal is 6 rolls for $4.50.

$1.80 $3.40$0.90 per roll $0.85 per roll

2 4

$4.50 $9.60$0.75 per roll $0.80 per roll

6 12

Explore

2. What methods did you use to determine which is the better deal?

3. Why would someone want to buy the more expensive option?

Some possible reasons why someone would buy the more expensive option: • they prefer a particular brand • quality of product• prefer shopping at a particular place• quantity they need

There are a number of ways to determine which of two prices are cheaper. Two of the most common methods are• Converting each to unit price• Converting each to a common number of units or a

common price

InformationIn the explore, the deals presented were examples of rate. Rate is a comparison of two amounts that are measured in different units. For example, km/h, $/L and m/s.

There are two ways in which rates can be compared:

1) Convert the rates so that they have the same number of units.

2) Determine the unit rate for each rate. A unit rate is a rate where the second term is 1. We usually use the key word per or the division symbol ( / ) to indicate a unit rate. For example: If a student earns $9.50 per hour, it is the same as $9.50/hour and means he earns $9.50 for every 1 hour of work.

Information

Method 1: Convert the rates so that they have the same number of units.

Method 2: Determine the unit rate for each rate.

$10.00 for 5 cupcakesvs.

$9.00 for 5 cupcakes

$2.00/cupcakevs.

$1.80/cupcake

For example, which is the better deal: $10.00 for 5 cupcakes or $18.00 for 10 cupcakes?

Example 1Determining the better buy

Determine which is a better buy.a) 10 bars of soap for $6.00 or 6 bars of soap for $3.90

b) a 12.5 oz bag of Doritos for $5.00 or a 3 oz bag for $1.05

c) a 5 gallon bucket of paint for $97.45 or a 1 gallon

bucket of paint for $21.95

Try this on your own first!!!!Try this on your own first!!!!

Example 1a: Solution

a) 10 bars of soap for $6.00 or 6 bars of soap for $3.90

Note: There are many ways to do these questions, but you should get the same answer…

$6.00 $3.90$0.60 / bar $0.65 / bar

10 6

The better buy is 10 bars of soap for $6.00.

In this example, we could determine the cost for 1 bar of soap and then compare this cost.

Example 1b: Solution

a 12.5 oz bag of Doritos for $5.00 or a 3 oz bag for $1.05In this example, we would likely want to find out the price per

ounce.


$5.00 $1.05$0.40 / oz $0.35 / oz

12.5 oz 3 oz

The better buy is a 3 oz bag for $1.05.

Example 1c: Solution

Since 5 is a multiple of 1, we can find out the price for 5 gallons of paint in each option.

Since the other option is $21.95 for 1 gallon, we can multiply both of these numbers by 5 to determine how much it costs for 5 gallons.

Since one option is $97.45 for 5 gallons of paint, we can leave this one.

$97.45 for 5 gallons


The better buy is 5 gallon bucket of paint for $97.45.

a 5 gallon bucket of paint for $97.45 or a 1 gallon bucket of paint for $21.95

$21.95 5 $109.75 / 5 gallons

Example 2Comparing Rates

Solve the following questions below.a) Natasha can buy a 26.4 lb turkey for $42.89 from the

butcher shop. The supermarket has turkeys on sale for $1.49/lb. Where should Natasha buy a turkey if she wants the best deal?

b) When Amelia uses a treadmill for 2 h, she burns 1400 Cal. When she uses an elliptical trainer for 30 min she burns 390 Cal. Who burns more calories per hour?

c) 50 L of oil costs $163. What is the cost of 20 L of oil?

Try this on your own first!!!!


Natasha can buy a 26.4 lb turkey for $42.89 from the butcher shop. The supermarket has turkeys on sale for $1.49/lb. Where should Natasha buy a turkey if she wants the best deal?

Since the supermarket price has a unit price of $1.49/lb, we only need to find the unit price of the turkey from the butcher shop.

The turkey from the supermarket is cheaper so it would be the best deal.

$42.89$1.62 / lb

26.4 lbs


When Amelia uses a treadmill for 2 h, she burns 1400 Cal. When she uses an elliptical trainer for 30 min she burns 390 Cal. Who burns more calories per hour?Find out how many calories each person burns in an hour.

Remember there are in 60 minutes in 1hr.

1400 cal 30min700 cal / h 0.5 h

2 h 60min/ h390 cal

780 cal / h0.5 h

Amelia burns more calories per hour using an elliptical trainer.

Example 2c: Solution

50 L of oil costs $163. What is the cost of 20 L of oil?

Now that we know how much 1 litre costs, we can easily multiply by 20 to find the cost of 20 L of oil.

The cost of 20 L of oil is $65.20.

$163$3.26 / L

50 L

$3.26 / L 20=$65.20

Example 3Comparing Rates Graphically

A hot tub contains 1600 L of water. Graph A represents the hot tub being filled at a constant rate. Graph B represents the hot tub being emptied at a constant rate.

a) Determine the rate of change represented in each graph.

b) Describe what the rate of change represents.



The rate of change corresponds with the slope of the graph.

rise 1600Lslope= = =20 L/min

run 80min

Filling a Hot Tub: Emptying a Hot Tub:

rise -1600 Lslope= = = -40L/min

run 40 min

When filling the hot tub, the rate of change is 20 L/min.When emptying the hot tub, the rate of change is -40 L/min.


The rate of change of 20 L/min indicates that the hot tub is being filled (positive rate of change) at a rate of 20 litres per minute.

The rate of change of -40 L/min indicates that the hot tub is being emptied (negative rate of change) at a rate of 40 litres per minute.

InformationRate of change is the rate at which one variable changes compared to another variable, where units are included.

Slope describes how two variables change in relation to each other. (Rise compared to run.)The rate of change can be determined using a graph and the slope formula. If two points and are on the graph of a line, then the slope of the line is

where the Greek symbol delta, , means change.

,A AA x y ,B BB x y

rise

slope = run

B A

B A

y yym

x x x

Information• A line that rises from left to right has a positive

slope, m > 0. • A line that falls from left to right has a negative

slope, m < 0. • A horizontal line has a slope of zero, m = 0. • A vertical line has an undefined slope, m

undefined.

Example 4Finding Slope

Haley is conducting a science experiment. She plotted the data shown on the right.a) Determine the rate of change, or slope, of each line segment.

i. From the graph, determine the rate of change or slope, m, for each line segment using the formula .

When moving from point A to point B, moving• upward is a positive rise • downward is a negative rise• right is a positive run• left is a negative run.

ii. Determine the slope, m, using the formula, .

rise

run

m

0 5 10 15 20 25 30 350

20

40

60

80

100

Water Temperature

Time (min)

Tem

per

atu

re (

˚C)

B A

B A

y ym

x x


Example 4Line Segment AB

Line Segment BC

rise

run0 0

10 r

0 C/min

ight 10

m

m

m

40 4 0010 10

0 C/ n0

mi

B A

B A

y ym

x x

m

m

0 5 10 15 20 25 30 350

20

40

60

80

100

Water Temperature

Time (min)

Tem

per

atu

re (

˚C)

rise

run30 down

15 ri

2 C/min

ght

3015

m

m

m

m

10 4025

3015

2 C/1

min0

B A

B A

y ym

x x

m

m

Example 4Line Segment CD

80up 8010

ris

right 1

e

run

0

8 C/min

m

m

m

90 10 8035 25 108 C/mi

n

B A

B A

m

ym

m

yx x

0 5 10 15 20 25 30 350

20

40

60

80

100

Water Temperature

Time (min)

Tem

per

atu

re (

˚C)

b) Describe what the rate of change means for each line segment. Is the line horizontal, does the line rise from left to right, or does the line fall from left to right?

i. line segment AB

ii. line segment BC

iii. line segment CD

The line falls from left to right. The temperature is decreasing by .

Example 4

A horizontal line means no change in temperature. 0 C/min

The line rises from left to right. The temperature is increasing by .

2 C/min

8 C/min

Example 5Finding the Slope Given Points

A line passes through the point A(6, 3) and B(1, 7). Find the slope of the line.


Helpful HintThe slope of a line passing through points A and B is

B A

B A

y ym

x x

Example 5Finding the Slope Given Points

A line passes through the point A(6, 3) and B(1, 7). Find the slope of the line.

7 ( 3) 101 6 52

B A

B A

yx

m

m

ym

x

Helpful HintThe slope of a line passing through points A and B is

B A

B A

y ym

x x

( , )A Ax y ( , )B Bx y

Example 6Finding the Slope of a Vertical Line

A vertical line passes through the point A(3, 2) and B (3, 4). Find the slope of the line.


5

5 x

y

B(3, 4)5

A(3, 2)5

Example 6Finding the Slope of a Vertical Line

A vertical line passes through the point A(3, 2) and B (3, 4). Find the slope of the line.

5

5 x

y

B(3, 4)5

A(3, 2)5

4 ( 2) 63 3 0

undefined

B A

B A

y

m

x

m

ym

x

( , )A Ax y ( , )B Bx y

Example 7Graphing a Rate of Change

Jake drives a delivery truck and is paid $16.00 an hour plus a $20.00 gas allowance at the beginning of each workday for using his own vehicle.a) Complete the table below to show the relationship

between hours worked and pay.

b) Label the graph, and then sketch Jake’s earnings.

c) Determine the slope of the graph.

d) What do you notice about the calculated slope and Jake’s rate of pay?


Example 7 a & b: SolutionGraphing a Rate of Change

a) Complete the table below to show the relationship between hours worked and pay.

b) Label the graph, and then sketch Jake’s earnings.

20 13211610084685236 148

+16 +16+16+16 +16 +16 +16

A

A

x

y

B

B

x

y

0 1 2 3 4 5 6 7 80

20

40

60

80

100

120

140

Hours

Pay

($)

Example 7 c & d: SolutionGraphing a Rate of Change

c) Determine the slope of the graph.

84 36 484 1 3

$16/hour

B A

B A

m

m

y ym

x x

d) What do you notice about the calculated slope and Jake’s rate of pay?

The slope and his hourly rate of pay are the same, $16 / hour.

Need to Know:

• Rate is a comparison of two amounts that are measured in different units, such as, km/h, $/L, m/s.

• There are two methods you can use to compare rates.1. Convert the rates so that they have the same number of units, or2. Determine the unit rate for each item. A unit rate is a rate in which the second term is 1.

• Rate of change is the rate at which one variable changes compared to another variable, where units are included. The rate of change can be determined using a graph and the slope formula.

Need to Know:•Slope describes how two variables change in relation to one another. If two points and are on a the graph of a line, then the slope of the line is

.

• Slope can be positive, negative, zero or undefined. • Lines with a positive slope rise from left to right.• Lines with a negative slope fall from left to the

right.• A line with a slope of 0 is horizontal.• A line with an undefined slope is vertical.

,A AA x y ,B BB x y

rise

slope = run

B A

B A

y yym

x x x

You’re ready! Try the homework from this section.

topic 1 slope and rate of change unit 8 topic 1. explore susan is shopping for the best buy on paper...

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