Chapter 1 Measurement · PDF fileChapter 1 Measurement Math 1201 3 Example 4: The perimeter of...

Chapter 1 Measurement Math 1201 1

Chapter 1 – Measurement

Sections 1.1-1.3:

Imperial units inch, foot, yard, mile

SI units metric system

(I) Converting between imperial units by unit analysis

Conversion Table:

(a) yards to feet

= yds x

Sometimes conversions may involve more than one scale

(b) miles to inches

= mi x x

1 ft. = 12 in.

1 yd. = 3 ft.

1 yd. = 36 in.

1 mi. = 1760 yd.

1 mi. = 5280 ft.

From the scale above, what units must be

placed in the numerator (top) and

denominator (bottom) so that yards cancel

and feet remain?

From the scale above, what units must be

placed in the numerators (top) and

denominators (bottom) so that miles

cancel and inches remain?


Relationships between units:

Example 1:

(a) Convert 100 yd to ft.

(b) Convert 100 yd to inches

Example 2: A sunken ship is discovered by sonar to be in 13 200 ft of water.

Convert this depth to miles.

Example 3: Convert 19.75 yds to yards, feet and inches.

1 ft. = 12 in.

1 yd. = 3 ft.

1 yd. = 36 in.

1 mi. = 1760 yd.

1 mi. = 5280 ft.


Example 4: The perimeter of a ceiling in a room is measured for the purpose of

installing crown molding. The perimeter of the ceiling is 272 in.

(a) What is the perimeter of the ceiling in feet?

(b) The cost of crown molding is $3.75/ft.. Determine the cost of ordering

crown molding to install in the room.

Example 5: There is 6 yd of material to be cut into strips that are 5in wide. How

many strips can be made?


Math 1201 questions page 11-12 of Textbook

7. Convert: a) 3 ft. to inches, b) 63 yd. to feet, c) 48in. to feet

8. Convert a) 2 mi. to feet b) 574 in. to yards, feet and inches

10. Carolyn is building a pen for her dog.

The perimeter of the pen is 52 feet.

a) Covert the perimeter to yards and feet.

b) The fencing material is sold by the yard. It costs $10.99/yd. What is the cost of

material before taxes?

11. David has 10 yd. of material that he will cut into strips 15in. wide to make mats.

a) How many mats can David make?

Answers: 7a) 36in b) 189ft c) 4ft

8.a) 10560 ft b) 15yd 2 ft 10 in

10a) 17yd 1ft b) $197.82

11.a) 24 mats

15.a) $119.99 b) $18.59

16. 1062 ft

18. 28 tulip bulbs


(II) Converting between SI units

The smallest metric measurement on the ruler below is the _____________.

How many divisions make up 1 cm? Answer:_______________

metric measurements for determining length is based on increments of 10.

SI units Abbreviation Relationship between

units

millimeter mm

centimetre cm 1 cm = 10 mm

metre m 1 m = 100 cm

kilometre km 1 km = 1000 m

Example 5: Convert each length to the indicated measurement.

1. 3 cm to mm 2. 15 m to km 3. 18 mm to cm

4. 360 m to cm 5. 6 km to m 6. 17 cm to m


(III) Converting Imperial Units to SI units

1 foot = 12 inches 1 yard = 3 feet 1 mile = 1760 yards

1 inch = 2.54 centimetres ≐ 2.5 centimetres 1 mile ≐ 1.6 kilometres

Example 6: Convert each measurement to the nearest tenth.

(a) 8 in to cm (b) 264 mi to km (c) 7 ft to cm

(d) 1 yd = ______ mm

(e) 6 ft. 7 in to cm (f) 4 yd. 2 ft. 2 in to m


Questions Page 22-23 of testbook

Answers: 4a) 40.6cm b) 1.2mt c) 4.6m d) 1.5km

e) 9.7km f) 50.8mm

5.a) 1 in b) 8 ft c) 11yd d) 93mi

6a) 55.9 cm b) 256.5cm c) 9.6m

7.a) i) 2ft 6in ii) 3 yd iii) 6mi

10. the odometer is accurate; 142 km is close to 87 mi

12. a) Michael

14. 144 sections of casing

15. 28 in


REVIEW OF SURFACE AREA

Area formulas: (𝒖𝒏𝒊𝒕𝒔)𝟐

Rectangle: 𝑨 = 𝒍 × 𝒘 Square: 𝑨 = 𝒍 × 𝒍 or 𝑨 = 𝒍𝟐

Circle: 𝑨 = 𝝅 ∙ 𝒓𝟐

Triangle: 𝑨 =𝒃𝒉

𝟐 or 𝑨 =

𝟏

𝟐𝒃𝒉 Pythagorean Theorem: 𝑎2 + 𝑏2 = 𝑐2

Net Diagram the diagram formed if all sides were unfolded

Find the surface area of the following:

1. Square/Rectangular Prism

(A)

(B)

10cm

10cm

10cm

10cm

6cm

8cm


NOTE:

2.

12 cm

8 cm

NOTE: the top/bottom are circles and

the curved surface is a rectangle

whose length is the circumference of

the circle


Section 1.4: Surface Area of Right Pyramids and Right Cones

Pyramids: are named or described by the shape of its base.

SA of pyramids = SA of base + SA of lateral sides (triangles)

NOTE if the base polygon of a pyramid is regular (all sides =) then all

lateral triangles are congruent

Example 1: Determine the surface area of the regular triangular  pyramid  (regular

tetrahedron)

Sketch the tetrahedron net diagram

13 yd


The Right Pyramid

A three dimensional object that has

Triangular faces (lateral faces)

Base is a polygon (many sides)

Point on top is called the apex

S represents slant height (the height

Of the triangular face)

h is the height of the pyramid (the

Perpendicular distance from the

Apex to the center of the base)

The Lateral Area, AL, is the area

Of the exposed triangles (surface area without the base)

Example 2: Determine the lateral area and the surface area of the square  pyramid

Sketch the net diagram


Example 3: Determine the slant height of a square pyramid given the surface area.

Example 4: Determine the surface area of a square pyramid given the height of the pyramid.

Height is 8m and base is 12m


Example 5: Determine the surface area of a rectangular pyramid.

Net Diagram:

Questions: Page 34-35 #4,5,8a,10,16b,13b,18


Extra practice: Surface Area of a pyramid

1. If the surface area of the square pyramid is 160.8 cm2 then determine the slant height.

2. Determine the surface area for each pyramid.

(A)

6 cm

6 cm

S

10 cm

10 cm

12 cm


(B)

12 cm

18 cm

15 cm


(C)

(D)

10 cm

13 cm

22 cm

15 cm

16 cm

13 cm

Answers: 1. 10.4 cm

2.(A) 360 cm2 (B) 717.6 cm2

(C) 652.8 cm2 (D) 709.5 cm2


Right Cones:

SA = lateral area + base

SA =

Example 6: Determine the surface area of the right cone.

8 cm

18cm


Example 7: Determining an unknown measure of a cone

SA = 2325 cm2 and radius r = 10 cm.

Determine the height h of the cone

Questions: Page 34-35 #6a7b,8b,11,15,16a

10cm

h


Section 1.5: Volume of Right Pyramids and Right Cones

Volume – the amount of space an object occupies

Capacity – the amount of material a container holds

--volume and capacity are measured in cubic units. (𝑢𝑛𝑖𝑡𝑠)3

(A) Right Square Prisms, Right Rectangular Prisms 𝑉 = 𝐴𝐵 × 𝐻

The Volume of a Rectangular Pyramid is one-third the volume of the Right

Rectangular Prism with same dimensions

2 cm

3 cm

5 cm

2 cm

3 cm

5 cm


(B) Right Cylinder

Similarly, the volume of a right cone is one third the volume of a right

cylinder with the same base and height.

5 cm

16cm

5 cm

16cm

Questions: Page 42-43 #4a,5b,6a,7b,8,9


Volume Examples where you may need to find a missing measure first:

Example 1: A right square pyramid has a base side length of 6.4 cm. Each

triangular face has two equal sides of length 8 cm. Determine the height and volume of the pyramid.

6.4 cm

6.4 cm

8cm

8cm


Example 2: Determine the volume for:

Example 3: The height of the right cone below is 10 cm and its volume is 377 cm3

Determine the diameter.

4 cm

5 cm

10 cm

Questions: Page 42-43 #,11,15,18bd


Section 1.6: Surface Area and Volume of a Sphere:

Sphere: Hemisphere:

Surface Area: 𝑆𝐴 = 4𝜋𝑟2 Surface Area: 𝑆𝐴 = 2𝜋𝑟2 + 𝜋𝑟2 = 3𝜋𝑟2

Volume: 𝑣 =4

3𝜋𝑟3 or 𝑣 =

4𝜋𝑟3

3 Volume: 𝑣 =

2

3𝜋𝑟3 or 𝑣 =

2𝜋𝑟3

3

Example 1: An official basketball has a radius of 12.5 cm and usually

has a leather covering. Approximately how much leather,

in cm2, is required to cover 12 official basketballs?

Example 2: The surface area of a softball is approximately 50.24 in2.

Determine the diameter of the softball.

A sphere is the set of points in

space that are the same fixed

distance from a fixed point,

which is the center.


Example 3: Find the surface area of the hemisphere that has a radius of 8.0cm

Example 4: A fitness ball when inflated with air has a circumference of

198 cm. Determine the volume of the fitness ball to the nearest tenth of a cm3.

Questions: Page 51 #3d,4a,5,8,11,18,19,20


Section 1.7 Solving Problems Involving Objects

Example 1:

(A) Find Surface Area (B) Find Volume

2 ft

4 ft


Example 2: Below is a sketch of a grain bin. If a farmer's grain truck can hold 560 cubic feet of barley, how many truckloads of barley are required to fill the bin?


Example 3: Determine the surface area and volume of a right cylinder with a right cone of the same height removed.

(A) Find Surface Area (B) Find Volume

Questions: Page 59-60 #3-11 (omit #4)


Formulas

Surface Area of a

Cylinder

SA = 2𝜋𝑟2 + 2𝜋𝑟ℎ

Surface Area of a

Cone

SA = 𝜋𝑟2 + 𝜋𝑟𝑠

Surface Area of a

Sphere

SA = 4𝜋𝑟2

Volume of a Sphere V =

4

3 𝜋𝑟3 or V =

4 𝜋𝑟3

3

Volume of a Cone V =

1

3 𝜋𝑟2ℎ or V =

𝜋𝑟2ℎ

3

Volume of a Pyramid V = 1

3 Ah or V =

𝐿𝑤ℎ

3

Imperial 1 ft. = 12 in. 1 yd. = 3 ft.

1 mi. = 1760 yd. Imperial to SI Units

1 in. = 2.54 cm ~ 2.5 cm 1 mi. ~ 1.6 km

Chapter 1 Measurement · PDF fileChapter 1 Measurement Math 1201 3 Example 4: The perimeter of...

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