1 Unit 7 Measurement General Outcome: • Develop spatial sense and proportional reasoning. Specific Outcomes: 7.1 Solve problems that involve linear measurement, using: o SI and imperial units of measure o estimation strategies o measurement strategies 7.2 Apply proportional reasoning to problems that involve conversions between SI and imperial units of measure. 7.3 Solve problems, using SI and imperial units, that involve the surface area and volume of 3-D objects, including: • right cones • right cylinders • right prisms • right pyramids • spheres Topics: • Converting Between Units (Outcome 7.1 & 7.2) Page 2 • Conversion Problems (Outcome 7.1 & 7.2) Page 11 • Surface Area of Right Pyramids (Outcomes 7.3) Page 18 & Right Cones • Volume of Right Pyramids (Outcome 7.3) Page 26 & Right Cones • Surface Area & Volume of Spheres (Outcome 7.3) Page 35 • Surface Area & Volume of (Outcome 7.3) Page 43 Composite Objects
Transcript
1
Unit 7 Measurement
General Outcome: • Develop spatial sense and proportional reasoning.
Specific Outcomes:
7.1 Solve problems that involve linear measurement, using:
o SI and imperial units of measure
o estimation strategies
o measurement strategies
7.2 Apply proportional reasoning to problems that involve conversions between SI and imperial
units of measure.
7.3 Solve problems, using SI and imperial units, that involve the surface area and volume of
3-D objects, including:
• right cones
• right cylinders
• right prisms
• right pyramids
• spheres
Topics:
• Converting Between Units (Outcome 7.1 & 7.2) Page 2
• Conversion Problems (Outcome 7.1 & 7.2) Page 11
• Surface Area of Right Pyramids (Outcomes 7.3) Page 18
& Right Cones
• Volume of Right Pyramids (Outcome 7.3) Page 26
& Right Cones
• Surface Area & Volume of Spheres (Outcome 7.3) Page 35
• Surface Area & Volume of (Outcome 7.3) Page 43
Composite Objects
2
Unit 7 Measurement
There are many different types of measurement to determine
lengths. What are all the ways you can think of to measure
length?
You will notice that there are two different systems to measure
different lengths.
• SI system of measurement (Le Systeme
International) which we know as the metric system
contains:
• Other countries use Imperial units which are :
3
Ex) For each of the following which unit of measurement
would you use, and then estimate the length.
a) Height of a desk
b) Length of the text book
c) Width of the classroom
d) Distance form SPA to the West Side Costco in
Edmonton
e) Thickness of 1 sheet of paper
4
Converting between Units:
Often we will be asked to change from one unit to another. To
do this we will use equivalent fractions and a conversions chart.
Imperial to Imperial SI to Imperial Imperial to SI
1 ft = 12 in 1 mm ≅ 0.0394 in 1 in = 2.54 cm
1 yd = 3 ft 1cm ≅ 0.3937 in 1 ft = 30.48 cm
1yd = 36 in 1 m ≅ 39.3701 in 1 ft = 0.3048 m
1 mi = 1760 yd 1 m ≅ 3.2808 ft 1 yd = 91.44 cm
1 mi = 5280 ft 1 km ≅ 0.6214 mi 1 yd = 0.9144m
1 mi ≅ 1.6093 km
1 in = 25.4mm
Ex) Convert each of the following
a) 12 cm to inches b) 36 mi to km
c) 5 ft and 10 in to m d) 5312 cm to yd
e) 101.6 cm to feet f) 632 in to yards with feet
and inches and inches
5
Ex) Find both of the missing sides in m
55˚
x 35 ft
y
6
Converting Between Units Assignment:
1) Which imperial unit is the most appropriate unit to measure each item?
a) The height of your desk b) the thickness of a mattress
c) The width of a car d) the length of flat panel TV
e) The length of a piece of f) the distance from the school
notebook paper to your home
g) The height of the classroom h) The length of your arm from
door your wrist to your elbow
2) Convert the following.
a) 2 miles to feet b) 574 inches to yards, feet, and
inches
c) 165 inches to yards, feet and d) 7390 feet to miles, yards, and
inches feet
7
3) Carolyn is building a pen for her dog. The perimeter of the pen is 52 ft.
a) Convert the perimeter to yards and feet.
b) The fencing material is sold by the yard. It costs $10.99 /yd. Determine the
cost of the material.
4) David has 10 yd. of material that he will cut into strips 15 in. wide to make
mats. Determine how many mats David can make.
5) In 2008, Sandy Allen and Leonid Stadnyk were the world’s tallest living
woman and man. Their respective heights are 7 ft. 7 in. and 8 ft. 5 in.
Determine how many inches shorter Sandy is when compared to Leonid.
8
6) Convert each measurement. Round each answer to the nearest tenth.
a) 16 in. to cm b) 4 ft. to m
c) 1650 yd. to km d) 6 mi. to km
e) 2 in. to mm f) 25 mm to in.
g) 2.5 m to ft. h) 10 m to yd.
i) 150 km to mi. j) 5 yd. to m
9
7) Convert the following. Round your answers to the nearest tenth.
a) 1 ft. 10 in. to cm
b) 2 yd. 2 ft. 5 in. to cm
c) 10 yd. 1 ft. 7 in. to m
8) Convert each measurement.
a) 75 cm to feet and nearest inch
b) 274 cm to yards, feet, and nearest inch
10
9) The dimensions of a lacrosse field are 110 yd. by 60 yd. What are these
dimensions to the nearest tenth of metre?
10) The Fraser River is approximately 1375 km long. The Tennessee River is
approximately 886 mi. long. Determine which river is longer?
11) On a road trip in Montana, Elise sees a sign that reads that Helena is 87 miles
away. To test the accuracy of her car’s odometer she tracks the distance she
drove from that sign to Helena’s city limit. Her odometer showed a distance of
142 km. Is the odometer accurate?
11
Conversion Problems:
Imperial to Imperial SI to Imperial Imperial to SI
1 ft = 12 in 1 mm ≅ 0.0394 in 1 in = 2.54 cm
1 yd = 3 ft 1cm ≅ 0.3937 in 1 ft = 30.48 cm
1yd = 36 in 1 m ≅ 39.3701 in 1 ft = 0.3048 m
1 mi = 1760 yd 1 m ≅ 3.2808 ft 1 yd = 91.44 cm
1 mi = 5280 ft 1 km ≅ 0.6214 mi 1 yd = 0.9144m
1 mi ≅ 1.6093 km
1 in = 25.4mm
Ex) For the following shape answer each part
a)
15in
20cm
i) What is the perimeter in feet
ii) What is the area in cm2
12
b) What is the perimeter of the triangle in yards?
8m
45˚
Ex) Bill wants to carpet his rectangular living room which has
dimensions of 525cm by 8.3 yd. If the carpet cost $2.25
per square foot, how much will it cost?
13
Ex) A map has a scale of 1cm:50km (1 cm on the map is
actually 50 km). If 2 cities are 11.2 cm apart on the map,
how many actual miles are they apart?
Ex) The fastest moving insect is the large tropical cockroach. It
scurries at speeds of up to 2.3 feet per second. How many
miles a roach can travel in 1.5 hours.
14
Conversion Problems Assignment:
1) A wallpaper border is to be pasted halfway up the wall around a bedroom.
12 ft. 9 in.
8 ft. 1 in.
2 ft. 6 in.
a) Determine the total length of the border needed.
b) Each roll of border is purchased in 12 ft. rolls and each roll sells for
$12.49. Determine the cost to border the bedroom.
2) In a basement renovation, the contractor measured the length of a wall in a
square room as 18 ft. 4 in. The width of the doorway is 3 ft. The contractor
plans to place wood trim along the bottom of each wall. The trim costs
$1.69 /ft. Determine the cost of the trim for the room.
15
3) A 3-D puzzle of the Eiffel Tower has a scale of 1:360. In the puzzle, the tower
is 2
355
in. tall. Determine the height of the Eiffel Tower in feet.
4) A map of Quebec has a scale of 1:1 500 000. On the map, the distance between
Trois-Rivieres and Quebec City is 5
28
in. Determine the distance between
these cities to the nearest mile.
5) A student can walk 30 ft. in 10 seconds. How far could she walk in 1 hour?
Express your answer in miles and yards.
16
6) Twenty reams of paper form a stack 40 in. high. Each ream costs $3.
Determine the value of a stack that has the same height as Mount Logan, which
is 19 500 ft. high.
7) A retail fabric store advertises a storewide sale. It lists a certain material for
$0.89 /yd. A fabric warehouse is selling the same fabric for $0.93 /m.
Determine which store has the better deal.
8) The tallest structure in Canada is the CN Tower in Toronto. It is 553.3 m tall.
The tallest structure in in United States is the Willis Tower in Chicago. It is
1451 ft. tall.
a) Determine the height of the CN Tower in feet and the height of the Willis
Tower in metres.
b) Which structure is taller? Determine the difference in heights in both
metres and feet.
17
9) The rim of a basketball net is mounted 10 ft. off the ground. A basketball
player has a maximum reach of 2.5 m. Determine how high, in inches, the
player needs to jump to reach 6 in. above the rim.
10) An electrician was hired to run the wires for a surround-sound stereo speaker
system. She purchased 2 rolls of 14-gauge wire. Each roll contains 4 m of wire.
For each of 2 speakers, 2 ft. of wire are required. For each of the other 2
speakers, 11 ft. of wire are required. Will the electrician have enough wire? If
your answer is no, what length of wire in centimetres will she need? If your
answer is yes, what length of wire in centimetres will be left over?
18
Surface Area of Right Pyramids & Right Cones:
A right pyramid is a 3-dimensional object that has triangular
faces and a base that is a polygon. The shape of the base
determines the name of the pyramid. The triangular faces meet
at a point called the Apex. The height of the pyramid is the
perpendicular distance from the apex to the center of the base.
When the base of the right pyramid is a regular polygon, the
triangular faces are congruent. Then the Slant Height of the
right pyramid is the height of the triangular face.
19
Ex) Determine the surface area of the right pyramid shown
below to the nearest square centimeter.
Ex) A right rectangular pyramid has base dimensions of 8 ft. by
10 ft. and a height of 16 ft. Determine the surface area of
the pyramid to the nearest square foot.
20
A Right Cone is a right pyramid with a circular base. The
surface area of a right cone is given by the formula:
SA =lateralheight
+ basearea
2SA rs r = +
where s = slant height
r = radius of base
Ex) A right cone has a base radius of 2 ft. and a height of 7 ft.
Determine the surface area of the cone to the nearest
square foot.
21
Ex) The lateral area of the cone is 220 cm2. The diameter of the
cone is 10 cm. Determine the height of the cone to the
nearest tenth of a cm.
Ex) The Great Pyramid of Giza has a square base with length
755 ft. and an original height of 481 ft. Determine its
original surface area to the nearest square foot. (Do not
include the base in the calculation.)
22
Ex) A farmer uploaded grain onto a tarp on the ground. The
grain formed a cone-shaped pile that had a diameter of
12 ft. and a height of 8 ft. Determine the surface area of the
exposed grain to the nearest square foot.
23
Surface Area of Right Pyramids & Right Cones Assignment:
1) Determine the surface area of each right pyramid and cone given below. Round
your answers to the nearest tenth if necessary.
a) Square Pyramid b) Regular Tetrahedron
c) d)
e) Right Square Pyramid f) Right Cone
24
g) h)
2) The slant height of a right square pyramid is 73 ft. and the side length of the
base is 48 ft. Determine the lateral area of the pyramid to the nearest square
foot.
3) Aiden built a cone-shaped volcano for a school science project. The volcano
has a base diameter of 32 cm and a slant height of 45 cm.
a) Determine the lateral area of the volcano to the nearest tenth of a square
centimeter.
b) The paint for the volcano’s surface costs $1.99 /jar, and one jar of paint
covers 400 cm2. Determine how much it will cost to paint the volcano.
25
4) Determine the indicated slant height for each figure shown below. Round your
answers to the nearest tenth of a unit.
a) Right Cone b) Right Square Pyramid
27012 mmSA = 265.5 mSA =
5) A right pyramid has a base that is a regular hexagon with side length 5.5 cm.
Each triangular face has 2 equal sides with length 7.5 cm. Determine the
surface area of this pyramid.
6) A right cone has a height of 8 ft. and a base circumference of 12 ft. Determine
the surface area of the cone to the nearest square foot.
26
Volume of Right Pyramids & Right Cones:
The volume of a Right Prism The volume of a Right Pyramid
is equal to the area of its base with the same base and height
times its height is 1
3 the volume of the right
prism.
V = area ofbase
height V =1
3 area of
base height
Volume of a right rectangular Volume of a right rectangular
prism pyramid
V lwh= 1
3V lwh=
27
Ex) Determine the volume of the right square pyramid to the
nearest cubic inch.
Ex) Determine the volume of a right rectangular pyramid with
base dimensions of 5.4 cm by 3.2 cm and a height of 8.1 cm.
Round your answer to the nearest tenth of a cubic cm.
28
Volume of a right cylinder Volume of a right cone
2V r h= 21
3V r h=
Ex) Determine the volume of the cone shown below to the
nearest cubic inch.
29
Ex) A cone has a height of 4 yd. and a volume of 205 cubic
yards. Determine the radius of the base of the cone to the
nearest yard.
Ex) A right square pyramid has a base side length of 3.5 m.
Each triangular face has two equal sides of length 4.5 m.
a) Sketch this pyramid b) Determine its height to
the nearest tenth of a meter.
c) Determine the volume of the pyramid to the nearest
tenth of a cubic meter.
30
Volume of Right Pyramids & Right Cones Assignment:
1) Determine the volume of each figure shown below. If necessary, round your
answers to the nearest tenth.
a) b)
c) d)
31
e) f)
g) h)
2) A regular tetrahedron has a base area of 68.0 m2 and height of 10.2 m. Sketch
the tetrahedron and determine its volume to the nearest tenth of a cubic metre.
32
3) A right cone has a slant height of 12 yd. and a base diameter of 4 yd. Sketch
the cone and determine its volume to the nearest tenth of a cubic yard.
4) A stone monument has the shape of a square pyramid. Its slant height is 1.6 m
and the side length of its base is 0.8 m. Determine the volume of the monument
to the nearest tenth of a cubic metre.
5) Determine the volume of a right rectangular pyramid with base dimensions of
6 ft. by 12 ft. For each triangular face, the equal sides have length 6 yd. Round
your answer to the nearest cubic foot.
33
6) Determine the measure of each indicated dimension. Round your answers to
the nearest tenth of a unit.
a) Right Rectangular Prism
388.8 cmV =
b) Right Square Pyramid
3554.9 mV =
34
c) Right Cylinder
3219.0 mV =
d) Right Cone
3164.9 cmV =
35
Surface Area & Volume of a Sphere:
Surface Area:
The surface area of a sphere is given by:
24SA r=
Ex) The diameter of a baseball is approximately 3 in.
Determine the surface area of a baseball to the nearest
square inch.
Ex) The surface area of a lacrosse ball is approximately 20
square inches. Determine the diameter of a lacrosse ball to
the nearest tenth of an inch.
36
Volume:
The surface area of a sphere is given by:
34
3V r=
Ex) The sun is an approximate sphere with a diameter of
870 000 miles. Determine the approximate volume of the
sun.
Ex) Determine the following for a hemisphere (half of a
sphere) with a radius of 8.0 cm.
a) Volume b) Surface Area
37
Ex) The surface area of a tennis ball is approximately 127 cm2.
Determine the radius of the tennis ball to the nearest tenth
of a cm.
Ex) A sphere has a diameter of 12 cm and a hemisphere has a
radius of 8 cm.
a) Which object has the greater volume?
b) Which object has the greater surface area?
38
Ex) Giselle has a block of wood that measures 14 cm by 12 cm
by 10 cm. She is making a wooden ball in tech class. What
percent of wood will be waisted?
Ex) A balloon with radius of 10 cm is blown up so that its
radius is 3 times larger. For the inflated balloon and the
original balloon
a) how do the circumferences compare?
b) how do the surface areas compare?
c) how do the volumes compare?
39
Ex) A hemisphere has a circumference of 47.1 m. Determine
the surface area and the volume of the hemisphere to the
nearest tenth of a unit.
40
Surface Area & Volume of a Sphere Assignment:
1) Determine the surface area and volume of each sphere given below. Round all
answers to the nearest tenth.
a) b)
c) d)
41
2) Determine the surface area and volume of each hemisphere given below.
Round all answers to the nearest tenth.
a) b)
3) A sphere has a radius of 8.4 m. Determine its surface area and volume to the
nearest tenth of a unit.
4) A sphere has a surface area of 452 square inches. Determine the diameter of
the sphere to the nearest inch.
42
5) A glass bowl approximates a hemisphere with diameter 20 cm.
a) Determine the capacity of the bowl to the nearest tenth of a litre.
( )31000 cm 1 L=
b) One cup is 250 mL. How many cups of punch can the bowl hold?
6) The centre of a doughnut is removed and formed to make a sphere of dough
with diameter 2.5 cm. A batch of these spheres is to be covered in a sugar
glaze. There is enough glaze to cover an area of 4710 cm2. Determine how
many spheres can be glazed.
7) A hemisphere has a circumference of 47.1 m. Determine the surface area and
volume of the hemisphere to the nearest tenth of a unit.
43
Surface Area & Volumes of Composite Objects:
Ex) Determine the volume of the following composite objects
given below.
a)
b)
44
Ex) Determine the surface area of the following composite
objects.
a)
b)
45
Ex) A tool shed is formed by a rectangular prism with a
triangular prism as its roof. Determine the surface area of
the tool shed to the nearest square foot.
Ex) A rocket has a cylindrical body and a cone-shaped nose.
The cylinder is 55 cm long with a radius of 6 cm. The cone
has a slant height of 12 cm and has the same radius as the
cylinder.
a) Determine the surface area of the rocket to the nearest
tenth of a square cm.
b) Determine the volume of the rocket to the nearest tenth
of a cubic cm.
46
Surface Area & Volume of Composite Objects Assignment:
1) Determine the surface area and volume of each composite object. If necessary,
round all answers to the nearest tenth of a square unit.
a)
b)
47
c)
d)
48
e)
f)
49
g) A Right Square Prism with a Right Square Pyramid Removed
h) A Right Cylinder with a Hemisphere Removed
50
2) Determine the measure of the indicated dimension for each object shown.
Round your answers to the nearest tenth of a unit.
a)
Curved 2219 inSA =
b)
Total 2137.2 cmSA =
51
3) Shown below are two different grain storage bins.
Bin A Bin B
a) Determine which storage bin holds more grain and state how much more
grain it holds when compared to the other bin.
b) Each storage bin has a cement base. The materials for the walls and roof of
the square-based bin cost $10.49 per square foot. The materials for the
walls and roof of the circular-based bin cost $9.25 per square foot.
Determine which bin is cheaper to build and state how much cheaper it is
when compared to the other bin.
52
Answers: Converting Between Units Assignment:
1. a) inches b) inches c) feet d) inches e) inches f) miles
g) feet h) inches
2. a) 10560 feet b) 15 yards, 2 feet, 10 inches c) 4 yards, 1 foot, 9 inches
d) 1 mile, 58 yards, 703 yards, 1 foot
3. a) 17 yd. 1 ft. b) $197.82
4. 24 mats
5. 10 in.
6. a) 40.6 cm b) 1.2 m c) 1.5 km d) 9.7 km e) 50.8 mm
f) 1.0 in. g) 8.2 ft. h) 10.9 yd. i) 93.2 mi. j) 4.6 m
7. a) 55.9 cm b) 165.1 cm c) 9.6 m
8. a) 2 ft. 6 in. b) 2 yd. 2 ft. 10 in.
9. 100.6 m by 54.9 m
10. 1375 km 854.4 mi. The Tennessee River is longer.
11. 142 km 88.2 mi. Her odometer is off by a bit.
Conversion Problems Assignment:
1. a) 39 ft. 2 in. b) 4 rolls for a total of $49.96
2. $119.99
3. 1062 ft.
4. 62 mi.
5. 2 mi. 80 yd.
6. $351 000
7. The warehouse is cheaper by $0.04 /m or $0.04 /yd.
8. a) CN Tower is 1815 ft., Willis Tower is 431.3 m
b) The CN Tower is taller by 364 ft. or 122 m
9. 28 in.
10. Yes there will be enough wire. There will be about 3.8 cm of wire left from
each roll.
Surface Area of Right Pyramids and Right Cones Assignment
1. a) 168 in2 b) 293.8 cm2 c) 150.8 in2 d) 2356.2 cm2 e) 896 cm2
f) 628.3 yd2 g) 87.3 m2 h) 176.3 ft2
2. 7008 ft2
53
3. a) 2261.9 cm2 b) $11.94
4. a) 69 mm b) 7.6 m
5. 193.7 cm2
6. 61 ft2
Volume of Right Pyramids & Right Cones Assignment:
1. a) 288 yd3 b) 1920 ft3 c) 1570.8 cm3 d) 804.2 m3
e) 18 m3 f) 168 yd3 g) 37.7 m3 h) 2948.9 cm3
2. 231.2 m3
3. 148.7 yd3
4. 0.3 m3
5. 401 ft3
6. a) 4.7 cm b) 10.5 m c) 3.3 m d) 7.4 cm
Surface Area & Volume of a Sphere Assignment:
1. a) 2314.2 cmSA =
3523.6 cmV = b) 232.2 mSA =
317.2 mV =
c) 2201.1 ftSA =
3268.1 ftV = d) 298.5 cmSA = 392.0 cmV =
2. a) 2339.3 mSA =
3452.4.6 mV = b) 2190.9 ydSA = 3190.9 ydV =
3. 2886.7 mSA =
32482.7 mV =
4. 12 in.d =
5. a) 4.2 LV = b) 16.8 cups
6. 240
7. 2706.1 mSA = 31764.5 mV =
Surface Area & Volume of Composite Objects Assignment:
1. a) 2169.6 cmSA = 3179.1 cmV = b) 21040 ftSA = 32100 ftV =
