Geometry and Measurement - Grade 12 College Mathsheldon12cmath.weebly.com/uploads/1/6/4/6/16463216/...Geometry and Measurement Key Ideas: •Unit Conversion: Metric and Imperial •Area

Unit 2 Measurement.notebook

1

February 14, 2013

Jan 129:00 AM

Geometry and Measurement

Key Ideas:

• Unit Conversion: Metric and Imperial• Area of Composite Figures • Volume• Surface Area • Optimization

Jan 1311:05 AM

Prefix Valuetera 1012 or 1 000 000 000 000giga 109 or 1 000 000 000mega 106 or 1 000 000kilo 103 or 1 000hecto 102 or 100deka 101 or 10

100 or 1deci 10-1 or 0.1centi 10-2 or 0.01milli 10-3 or 0.001micro 10-6 or 0.000 001nano 10-9 or 0.000 000 001pico 10-12 or 0.000 000 000 001

The Metric System

When you convert a larger unit to a smaller unit, you MULTIPLY the smaller number by the difference in the 10n exponents

∴ the number get will get BIGGER → move to the RIGHT.

When you convert a smaller unit to a larger unit, you DIVIDE the larger number by the difference in the 10n exponents.

∴ the number get will get SMALLER → move to the LEFT.


2

February 14, 2013

Jan 1311:28 AM

4.5 L to mL 12 km to cm

27 000 mL to L 69 500 000 mg to kg

Examples:

big to small > move right

how many units? 3

left or right?

4500 mL

how many units?

left or right?

5

big to small > move right

1,200,000 cm

small to big > move left

how many units? 3

left or right?

27 L

how many units? 6

left or right?

69.5 Kg

small to big > move left

Convert the following to the given units.

Jan 1311:38 AM

Length Mass Volume

1 ft = 12 inches 1 lb = 16 ounces 1 gal = 4 qt 1 yd = 3 ft 1 Ton = 2000 lb 1 qt = 2 pt1 mile = 5280 ft1 pt = 16 fluid oz

The Imperial System

When you convert a larger unit to a smaller unit, you MULTIPLY the number by the conversion factor. THE NUMBER SHOULD GET BIGGER!!!

When you convert a smaller unit to a larger unit, you DIVIDE the number by the conversion factor. THE NUMBER SHOULD GET SMALLER!!!


3

February 14, 2013

Jan 1311:56 AM

3.2 ft to inches 0.75 lbs to ounces

2456 ft to yd 12.5 qt to gal

Examples: Convert the following to the given units.

conversion factor:

multiply or divide?

3.2 x 12 = 38.4"

12

multiply

conversion factor:

multiply or divide?

0.75 x 16 = 12 oz

16

multiply

conversion factor:

multiply or divide?

2456 ÷ 3 = 818.7 yd

3divide

conversion factor:

multiply or divide?

12.5 ÷ 4 = 3.2 gal

4divide

Jan 1312:16 PM

Converng between Metric and Imperial

Imperial to Metric Metric to Imperial

1 inch = 2.54 cm 1 cm ≈ 0.3937 inch

1 foot = 30.48 cm 1 m ≈ 39.37 inches

1 foot = 0.3048 m 1m ≈ 3.2808 feet

1 mile ≈ 1.609 km 1 km ≈ 0.6214 mile

Metric and Imperial Units of Length

Use conversion factors to change imperial units to metric units, or vice versa.

The symbol ' represents feet and the symbol " represents inches.

Imperial to Metric Metric to Imperial

1 fluid ounce = 28.413 mL 1 mL = 0.0352 fluid ounce

1 pint = 0.568 L 1 L = 1.7598 pints

1 quart = 1.1365 L 1L = 08799 quart

1 gallon = 4.546 L 1 L = 0.22 gallon

Metric and Imperial Units of Capacity

Capacity is a measure of how much liquid a container can old. Use conversion factors to change imperial units to metric units, or vice versa..

1 mL = 1 cm3


4

February 14, 2013

Jan 131:04 PM

Recall:

• If you SQUARE your units, you must SQUARE your conversion factor!

• If you CUBE your units, you must CUBE your conversion factor!

i.e. 1 ft = 12" 1 ft2 = 144 in2

i.e. 1 m = 100 cm 1 m3 = 1,000,000 cm3

conversion factor: conversion factor:


71 Km2 to m2 3.7 yd3 to ft3

1 Km = 1000 m

1 Km2 = 1,000,000 m2

Bigger or Smaller? Bigger

71 x 1,000,000 = 71,000,000 m2

1 yd = 3 ft

1 yd3 = 27 ft3

Bigger or Smaller? Bigger

3.7 x 27 = 99.9 ft3

Jan 1312:39 PM

Remember to ask yourself: "should the number get bigger or smaller?"

How do you know? Look to your units!

• If you are converting from a bigger unit to a smaller unit the number should get BIGGER!

• If you are converting from a smaller unit to a bigger unit, the number should get SMALLER!

Steps for Conversion

Step 1: Find the conversion factor by locating the appropriate chart.

Step 2: MULTIPLY the given number by the conversion factor.

• If the units are squared, square (x2) the conversion factor.• If the units are cubed, cube (x3) the conversion factor.


5

February 14, 2013

Jan 1312:45 PM

2.4" to cm 325 mL to ounces

5 Km to miles 2.8 Kg to lbs.

smaller or bigger?

2.4 x 2.54 = 6.096 cm

2.54

bigger smaller or bigger? 325 x 0.0352 = 11.0 oz

0.0352

smaller

smaller or bigger?

5 x 0.6214 = 3.1 miles

0.6214smaller


conversion factor: conversion factor:

conversion factor:

Jan 138:01 PM

Volume: The amount of 3dimensional space occupied by an object.

Volume of a prism: Area of Base x Height

Do not rely on your formula page. You must practice using the formulas.

Example: Find the volume of the following cylinder with a diameter of 4 cm and a height of 6 cm.

V = area of base x heightV = πr2 x hV = π(2cm)2 x (6 cm)V = 75.4 cm3

Recall: For volume the units should always be cubed.


6

February 14, 2013

Jan 138:01 PM

Surface Area: The total area of the surface of an object.

DON'T use the 3D side of your formula page. Instead, use the AREA (2D) side of the formula page to find the AREA of EACH SIDE separately and then ADD them together.

Example: Find the surface area of the following cylinder with a diameter of 5 cm and a height of 9.6 cm.

Recall: For volume the units should always be cubed.

2.5 cm

9.6 cm

• the curved side of a cone and cylinder = LATERAL SURFACE (3D Side)• use the 3D formula for a SPHERE.

Exceptions:

Acircle = πr2

Acircle = π(2.5)2

Top and Bottom Curved Surface

Alateral surface = 2πrhAlateral surface = 2π(2.5)(9.6)

Acircle = 19.6 cm2 Alateral surface = 150.8 cm2

Total Surface Area = 2Acircle + Alateral surface

= 2(19.6 cm2) + 150.8 cm2

= 190 cm2

Sep 257:02 PM

Bell Work

Determine the surface area of the following cone in cm 2.

1'3"

10"

Solution:

= 12 in + 3 in= 15 in Acircle = πr2

Bottom:

Acircle = π(10)2

Acircle = 314.2 in2

Side:Alateral surface = πrs

s2 = 152 + 102s2 = 225 + 100s2 = 325s = √325s = 18 in

Alateral surface = π(10)(18)Alateral surface = 565.5 in2

Total Surface Area = 314.2 in2 + 565.5 in2

= 879.7 in2

Conversion: Imperial to Metric

1 inch = 2.54 cm ⇒ 1 in2 = 6.45 cm2 879.7 x 6.45 = 5,674.1 cm2


7

February 14, 2013

Jan 132:38 PM

2D Composite Figures

A figure that is made up from other simpler figures is called a composite figure.

To determine the area of a composite figure:

• Break it into simpler figure for which you know how to calculate the area.• Calculate the area of each part.• Add the areas.• Subtract the areas of any part removed from the figure.

Sep 255:46 PM

You obtain a summer job mowing lawns. A person with an odd shaped lot shown below hires you to cut his lawn. Given the dimension on the diagram, what area of lawn will you be cutting? State your answer in m2.

AREA OF THE RECTANGLE: ARectangle = l x w = 8' x 5' = 40 ft2

AREA OF THE TRIANGLE: ATriangle = b x h = 4' x 3' = 6 ft2 2 2

AREA OF THE LAWN: ALawn = 40 ft2 6 ft2 = 34 ft2

CONVERSION: 1 ft = 0.3048 m 1 ft2 = 0.0929 m2 34 ft2 = 3.2 m2

Example 1:


8

February 14, 2013

Sep 255:50 PM

Example 2: Using Trigonometry to Determine an Unknown Length

Carpenters have constructed the frame for a house and will nail pressboardover the frame. Determine the area of pressboard they need for the back wall of the house.

Solution:

Alarge rectangle = 5.75 m x 8.53 m = 49.0475 m2

Asmall rectangle = 81.5 in x 33.5 in = 2730.25 in2

1 in = 0.0254 m 1 in2 = 0.0006452 m2

∴ 2730.25 in2 = 2730.25 x 0.0006452 = 1.7616 m2

Atriangle = 8.53 m x ? 2

What is the height?

8.53 m ÷ 2

o

a

tan 220 = h4.265

h = 4.265tan220

h = 1.72 m

∴ Atriangle = 8.53 m x 1.72 m ÷2= 7.3358 m2

Surface Area = 49.0475 m2 1.7616 m2 + 7.3358 m2

= 54.6217 m2

∴ They need about 54.6217 m2 of pressboard for this side of the house.

Sep 255:50 PM

Bell Work: Determine the area of the composite Figure.

Solution:

A2 = (25 cm + 40 cm) x 30 cm = 975 cm 2

2

A1 = 40 cm x 30 cm = 1200 cm 21

2Total Area = 1200 cm2 + 975 cm2 = 2175 cm2


9

February 14, 2013

Sep 255:51 PM

3D Composite Objects

When a structure or object is made up from several simple objects, it is called a composite object.

To determine the volume of a composite object:

• Break it into simpler figure for which you know how to calculate the volume.• Calculate the volume of each part of the object.• Add the volumes.• Subtract the volumes of any parts that were removed.

Sometimes, the composite object can be viewed as a prism whose base is a composite figure. In these cases, you can use the formula: V = base area x height

Sep 257:31 PM

Example: Finding the Volume of a Composite Object

VRectangular Prism = 289.5 cm x 310.0 cm x 202.0 cm = 18,128,490.0 cm 3

VTriangular Prism = (310.0 cm x 79.0 cm) ÷2 x 289.5 cm = 3,544,927.5 cm3

Total Volume = 18,128,490.0 cm 3 + 3,544,927.5 cm 3

= 21,673,417.5 cm 3

1 m = 100 cm 1 m3 = 1,000,000 cm3 21,673,417.5 cm 3 = 21.7 m3

∴ The volume of the shed is approximately 18.2 m2.


10

February 14, 2013

Sep 257:31 PM

To determine the surface area of a composite object:

When you determine the surface area of a composite object, include only those faces that are face of the composite object. That is, the faces that are part of the surface of the object.

Example: Finding the Surface Area of a Composite Object

Afloor = 310.0 cm x 289.5 cm = 89,745 cm2

NOTE: There are 7 sides

Aside = 289.5 cm x 202.0 cm = 58,479 cm2

Afront/back = Arectangle + Atriangle

= (310.0 cm x 202.0 cm) + [(310.0 cm x 79.0 cm) ÷ 2]= 62,620 cm2 + 12,245 cm2

= 74,865 cm2

310.0 cm ÷ 2

155.0 cm

79.0 cmx

x

x2 = 155.02 + 79.02

x2 = 24,025 + 6,241

x2 = 30,266x = √30,266x = 174.0 cm

Aroof = 289.5 cm x 174.0 cm = 50,286 cm2

Total Surface Area = Afloor + 2Aside + 2Afront + 2Aroof

= 89,745+ 2(58,479) + 2(74,865) + 2(50,286)

= 89,745+ 116,958 + 149,730+ 100,572

=457,005.0 cm2

∴ The surface area of the shed is 457,005.0 cm2.

Sep 273:06 PM

MidChapter Review

• Begin "cheat sheet"

• p.86 ALL QUESTIONS

• Test Part 1 tomorrow!


11

February 14, 2013

Sep 273:07 PM

2.6 Optimizing Area & Perimeter

Length Width Area

Given the perimeter of a rectangle what dimensions (length and width) will maximize the area?

Investigation:

A park worker has 32 m of fencing to build a rectangular pen for rabbits. What is the maximum area that she can provide for the rabbits?

∴ An 8m by 8m SQUARE will maximize the area.

Sep 273:15 PM

Example:

Given the PERIMETER of a rectangle, a SQAURE will maximize the AREA.

LENGTH = WIDTH = AREA ÷ 4

What are the dimensions of a rectangle with perimeter 20 m and the maximum area? What is the maximum area?

Solution: Length = Width = 20 m ÷ 4 = 5 m

Area = 5m x 5m = 25m2

∴ A 5m by 5m square will create a maximum area of 25m2.


12

February 14, 2013

Oct 21:38 PM

Length Width Perimeter

Given the area of a rectangle what dimensions (length and width) will minimize the perimeter?

Investigation:

Jenny plans on building a 36m2 rectangular vegetable garden in her back yard. What is the minimum amount of material required to form the perimeter?

∴ An 6m by 6m SQUARE will minimize the perimeter.

Oct 21:51 PM

Example:

Given the AREA of a rectangle, a SQAURE will minimize the PERIMETER.

LENGTH = WIDTH = √AREA_______

What are the dimensions of a rectangle with area 45 m2 and the minimum perimeter? What is the minimum perimeter?

Solution: Length = Width = √45m2 = 6.71 m

Perimeter = 4 x 6.71m = 26.84 m

∴ A 6.71m by 6.71m square will create a minimum perimeter of 26.84m.


13

February 14, 2013

Oct 21:36 PM

Optimizing with Restrictions

There may be restrictions on the rectangle you are optimizing:

• The length and width may have to be whole number; or• The length and width may have to be multiples of a given number.

In these cases, it may not be possible to form a square. The maximum area or minimum perimeter occurs when the length and width are closest in value.

Sometimes one or more sides of the area to be enclosed are bordered by a wall or other physical barrier. In these cases, the optimal rectangle will not be a square. You can use diagrams or a table and graph to find the dimensions of the optimal rectangle.

Oct 22:05 PM

Enclosing NonRectangular AreasA hobby farmer is creating a fences exercise yard for her horses. She as 900m of flexible fencing and wishes to maximize the area. She is going to fence a rectangular or a circular area. Determine which figure encloses the greater area.

Solution:

Rectangular Area Circular Area

length = width = 900m ÷ 4 = 225 mArea = 225 m x 225 m = 50,625m2

The greatest rectangular area that can be enclosed is 50,625m2.

C = πd

perimeter = circumference

d = C ÷ π d = 900m ÷ π = 286.49m r = 286.49m ÷ 2 = 143.2m A = πr2 = π(143.2 m)2 = 64,422.3m2

The greatest circular area that can be enclosed is 64,422.3m2.

∴ The circular pen encloses the greater area.


14

February 14, 2013

Sep 273:07 PM

Bell Work

What are the dimensions of a rectangle with perimeter 60 m and the maximum area? What is the maximum area?

1)

What are the dimensions of a rectangle with area 49 m2 and the minimum perimeter? What is the minimum perimeter?

2)

length = width = 60m ÷ 4 = 15mArea = 15m x 15m = 225m2

length = width = √49m2 = 7m

Perimeter = 7m x 4 = 28 m

What is the area of a circle with a perimeter of 75 m?3)

d = 75m ÷ π = 23.87mr = 23.87m ÷ 2 = 11.9mA = π(23.87m)2 = 1,790m2

Sep 273:16 PM

2.7 Optimizing Volume & Surface Area

Given the SURFACE AREA of a rectangular prism, a CUBE will MAXIMIZE THE VOLUME.

Area of each side = SA ÷ 6 length = width = height = √Aside

Given the VOLUME of a rectangular prism, a CUBE will MINIMIZE THE SURFACE AREA.

length = width = height = ∛Volume


15

February 14, 2013

Sep 273:16 PM

Example 1:Rosa contracts a rectangular prism using exactly 384 square inches of cardboard. It has the greatest volume possible. What are the dimensions of the prism? What is the volume?

Solution:Aside = 384 in2 ÷ 6 = 64 in2

length = width = height = √64 in2 = 8 inVolume = 8 in x 8 in x 8 in = 512 in3

Example 2:Liam constructs a rectangular prism with a volume of exactly 1331 m3. It has the least surface area possible. What are the dimensions of the prism? What is its surface area?

Solution:length = width = height = ∛1331 m3 = 11 m

Aside = 11m x 11m = 121m 2

Surface Area = 6 x Aside = 6 x 121m2 = 7262

Oct 23:00 PM

There may be constraints on the prism you are optimizing:

• The dimensions may have to be whole number; or• The dimensions may have to be multiples of a given number.

In these cases, it may not be possible to form a cube. The maximum volume or minimum surface area occurs when the dimensions are closest in value.

Sometimes one or more sides of the object are missing or bordered by a wall or other physical barrier. In these cases, the optimal rectangular prism will not be a cube. You can use diagrams or a table and graph to find the dimensions of the optimal rectangular prism.

Optimizing with Constraints


16

February 14, 2013

Oct 23:03 PM

Optimizing Other ObjectsBen is designing a can with volume 350 mL. What is the minimum surface area of the can? Determine the dimensions of a can with the minimum surface area.

V = πr2h = πr22r = 2πr3

1 mL = 1 cm3 Optimized when the diameter equals the height.

r3 = V ÷ (2π) r = ∛r3

350 mL = 350 cm3

r3 = 350 cm3 ÷ (2π) = 55.7 cm3

r = ∛55.7 cm3 = 3.8 cm

h = 2r

h = 2 x 3.8 cm = 7.6 cm

SA = 2πr2 + 2πrh = 2π(3.8)2 + 2π(3.8)(7.6) = 272.2 cm2

Jan 129:00 AM

Key Ideas: • Unit Conversion• Area of Composite Figures • Volume• Surface Area • Optimization

Review: Test Tomorrow

To do: • Finish "cheat sheet"• Have Ms. Sheldon "signoff" on "cheat sheet" • Review Quiz & Journal for common errors• p.3841 #2ab, 9, 11, 17, 18a


17

February 14, 2013

Jan 132:04 PM

Using conversions to compare priceYou are almost out of gas and desperately need to fill up. You come across two gasstations. The first one is selling their gas at a price of $0.93/L. The second is selling theirgas at $2/gal. Which gas station should you choose?

Conversion factor: 1 US gal = 3.75 L

$2/gal = $2/3.75L = $0.53/L

REMEMER: JUST SUBSTITUTE AND DIVIDE!

∴ The second gas station has the better price.

Only convert one!

Documents

Geometry and Measurement - Grade 12 College Mathsheldon12cmath.weebly.com/uploads/1/6/4/6/16463216/...Geometry and Measurement Key Ideas: •Unit Conversion: Metric and Imperial •Area