B
Un+7
Practise
For help with questions I and 2, see Exampies I and 2.
c
1. For each composite figure,
• solve for any unknown lengths
• determine the perimeter
II,
Round to the nearest unit, if necessary.
a)
b)
13m
Gm
Scm
A
Scm
I
12m
5 cm
e)
5cm
----z
*‘
Sm
lam
1
d)
c)
cm
15cm
S cm
6 cm
2. Calculate the area of each composite figure.
Round to the nearest square unit, if necessary.
a)
b)
Cam
lOm
30mm
c)
d)
6cm
cm
4cm
S cm
e)
1)
10cm
ao cm
t
432 MHR’ChapterS
connect and Apply
3. a) What length of fencing is needed to surround this yard, to the
nearest metre?
b) What is the area of the yard?
c) Explain the steps you took to solve this problem.
iBm
urn
16m
4.
patrick is planning a garage sale. I-Is is painting six arrow signs to
direct people to his sale.
a) Calculate the area of one side of one arrow.
b) Each can of paint can cover 2 m2. How many cans of paint will
Patrick need to paint all six signs?
c) If the paint costs $3.95 per can, plus 8% PST and 7% CST, how
much will it cost Patrick to paint the six signs?
10 cm
75 cm
60cm;
20cm
5. Arif has designed a logo of her initial as shown. Use a ruler to make
the appropriate measurements and calculate the area of the initial, to
the nearest hundred square millimetres.
6. Create your own initial logo similar to the one in question
Calculate the total area of your logo.
5.
7. Use Technology
a) Use The Geometer Sketchpad® to draw your design from
question 0.
b) Use the measurement feature of The Geometer’s Sketchpad®
to measure the area of your design.
8. Chapter Problem One of the gardens Emily is designing is made
up of two congruent parallelograms.
a) A plant is to be placed every 20 cm around the perimeter
of the garden. Determine the number of plants Emily needs.
b) Calculate the area of her garden.
Sm
9. Use Technology Use The Geometer’s Sketchpad® to create a
composite figure made up of at least three different shapes.
a) Estimate the perimeter and area of the figure you created.
b) Determine the area using the measurement feature of
The Geometer’s Sketchpad®. Was your estimate reasonable?
8.2 Perimeter and Area of Composite Figures MHR 433
1!
10. An archery target has a diameter of
80 cm. It contains a circle in the centre
with a radius of 8 cm and four additional
concentric rings each 8 cm wide.
a) Find the area of the outer ring, to
the nearest square centimetre.
b) What percent of the total area is
the outer ring?
¶1
11. The area of a square patio is 5 m2.
a) Find the length of one of its sides, to the nearest tenth of a metre.
b) Find the perimeter of the patio, to the nearest metre.
12. Brandon works as a carpenter. He is framing a
rectangular window that measures 1.5 m by 1 m.
The frame is 10cm wide and is made up of four
trapezoids. Find the total area of the frame, to the
nearest square centimetre.
,
! Achievement Check
•
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Rp,esontin
SeIectInB marc
IN
Ocrc.a ScW,i
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13. Susan is replacing the shingles on her roof. The roof is made up of a
horizontal rectangle on top and steeply sloping trapezoids on each
side. Each trapezoid has a (slant) height of 4.5 m. The dimensions of
the roof are shown in the top view.
2am
Crsaa rif
18 m
em
11
cu
N
ii
.4
I
434 MHR Chapter 8
a) Calculate the area of the roof.
b) A package of shingles covers 10 in2. How many packages
will Susan need to shingle the entire roof?
c) Describe an appropriate way to round the number of packages
in part b).
xten d
14. Sanjav is designing a square lawn to fit inside a square yard wi (Ii
side length 10 m so that there is a triangular flower bed at each
corner.
a) Find the area of Sanjay’s lawn.
b) 1-low does the area of the lawn compare to the area of the flower
beds?
c) Sanjay’s design is an example of a square inscithod within
a square. The vertices of the inside square touch the sides of
the outside square but do not intersect. WiJIl your answer in
part b) always be true when a square is inscribed vi thin a square?
4
N
>lom
iOn
Explain
15. How does doubling (he radius of a circle affect its area? Justify your
answer using algebra.
16. Leonardo of Pisa lived in the 13th century in l’isa. Italy. He was
given the nickname Fibonacci because his fathers name x’as
Bonacci. Among his mathematical explorations is the sequence of
numbers 1, 1, 2, 3, 5, 8, 13. 21
a) Del ermine (he pattern rule for this sequence. and list the next
four terms.
b) Construct rectangles using consecutive terms for (lie sides.
The first rectangle is 1 by 1, the second is 1 by 2, the third is
2 by 3, and so on. Find the area of each rectangle.
c) Explore the ratios of the sides of the rectangles. Make
about this ratio.
conject tires
d) Explore the ratios of the areas of the rectangles. Make conjectures
about this ratio.
17.
Math Contest Determine the ratio of the perimeter of the smallest
square to the perimeter of the largest square.
4cm
30cm
18. Math Contest The midpoints of the sides of a rectangle that measures
10 cm by 8 cm are joined. Determine the area of the shaded region.
Scm
10cm
8.2 Perimeter and Area of Composite Figures’ MHR 435
fri
practise
For help with question 1, see Example 1.
1. Determine Lhe surface area of each object.
b)
a)
12.m4
8.5
8.5 cm
/
For help with question 2, see Exam pie 2
nearest
2. Determine the volume of each object. Round to the
cubic unit, when necessary.
b)
/\
a)
7N.
/L -\ :
15 mmj
2.6 m
-
j//
I’
20mm
\
\\
For help ii’ith questions 3 to 5, see Example 3
3. Determine the surface area of each object
b)
a)
V.
8 mm
5
10mm
cm
8 cm
4. Determine the volume of each object.
b)
m
a)
/1
j
1OCJ
/6cm
8 cm
2 m, and height 4
5. A rectangular prism has length 3 m, width
a) Determine the surface area of the prism
b) Determine the volume of the prism.
in.
Connect and Apply
If its length is 20 cm
6. A cereal box has a volume of 3000 cm3.
and its width is 5 cm, what is its height?
ids’ MHR 441
8.3 Surface Area and Volume of Prisms and Pyram
7. Sneferu’s North Pyramid at Dahshur, Egypt, is shown.
Its square base has side length 220 m and its height is 105 m.
a) Determine the volume of this famous pyramid.
b) Determine its surface area, to the nearest square metre.
p
8.The Pyramid of Rhafre at Giza, Egypt, was built by
the Pharaoh Khafre, who ruled Egypt for 26 years.
The square base of this pyramid has side length
215 m and its volume is 2 211 096 m3.
Calculate its height, to the nearest tenth
of a metre
9.
The milk pitcher shown is a right prism. The base has
an area of 40 cm2 and the height of the pitcher is 26 cm.
Will the pitcher hold 1 L of milk?
10. A juice container is a right prism with a base area of 100 cm2.
a) If the container can hold 3 L of juice, what is its height?
b) Describe any assumptions you have made.
11. Adam has built a garden shed in the shape
shown.
a) Calculate the volume of the shed, to the
2m
nearest cubic metre
b) Adam plans to paint the outside of the shed,
m
including the roof but not the floor. One can
4m
of paint covers 4 m2. How many cans
of paint wilt Adam need?
c) If one can of paint costs S16.95, what is the total cost, including
7% CST and 8% PST?
ia.
I
Chapter Problem The diagram shows the side view of the swimming
pool in Emily’s customer’s yard. The pooi is 4 m wide.
12m
t
1m________
3m
4
I
—3m—
a) Estimate how many litres of water the pooi can hold.
b) Calculate how many litres of water the pool can hold.
c) When the pool construction is complete, Emily orders water to.
fill it up. The water tanker can fill the pool at a rate of 100
How long will it take to fill the pool at this rate?
442 MHR’Chapter8
I
shorter
t is a right triangle with
tha
se
ba
a
s
ha
sm
pri
is 10 cm.
13. A triangular
e height of the prism
Th
.
cm
8
d
an
cm
6
re
sides that measu
prism.
t affects the volume of the
igh
he
the
g
lin
ub
do
w
a) Predict ho
original
lating the volume of the
lcu
ca
by
n
tio
dic
pre
b) Check your
the new prism.
prism and the volume of
curate?
Was your prediction ac
result.
ral? If so, summarize the
ne
ge
in
e
tru
s
tin
Is
d)
e
Achievement Check
8000 cm3 of rice. On
t containers that hold
en
fer
dif
o
tw
n
r.
sig
de
De
lin
cy
a
14. a)
prism and one should be
should be a rectangular
nearest square
area of each one, to the
ce
rfa
su
the
ne
mi
ter
b) De
centimetre.
nufacturer
recommend to the ma
u
yo
uld
wo
e
ap
sh
h
c) Whic
and wily?
ing
Seasoong ant Prov
—
SeocoLr Tools
Rep50000ire
Proolern Solvire
PrtlecOrnc
Cornecolng
Extend
ses. If their
ve congruent square ba
ha
sm
pri
a
d
an
id
ram
are? Explain.
15. A py
do their heights comp
w
ho
e,
sam
the
are
volumes
ramid. The frustum
d on a frustum of a py
ce
pla
be
to
is
tue
sta
16. A
been removed by
er the top portion has
aft
ing
ain
rem
rt
pa
is the
id.
to the base of the pyram
making a cut parallel
ce area of the frustum.
a) Determine the surfa
gilt paint that
4m
ing the frustum with
int
pa
of
st
co
the
e
the
lat
of
m
tto
bo
b) Calcu
the
int
pa
not necessary to
costs $49.50/rn2. It is
frustum.
sm is
a of a rectangular pri
are
ce
rfa
su
the
for
la
17. A formu
.
£4 = 2(]w ± vh ± Hi)
algebraically
sions is doubled. Show
en
dim
the
of
ch
ea
a) Suppose
is affected.
how the surface area
is doubled?
each of the dimensions
if
ted
ec
aff
e
lum
vo
b) How is the
ebraically.
Justify your answer alg
g together
cube is made by gluein
en
od
wo
ge
lar
A
st
ge cube along
1 8. Math Conte
right through the lar
de
ma
are
ts
Cu
s.
216 small cube
es.
ee perpendicular fac
the diagonals of thr
t?
all cubes remain uncu
How many of the sm
CommuncatLr
3m
4m
Ws• MHR 443
e of Prisms and Pyram
lum
Vo
and
ea
Ar
8.3 surface
communicate Your Understanding
A cone is formed from a circle with a 90°
a, sector
removed. Another cone is formed
from a semicircle with the same radius.
How do the two cones differ? How are
they the same?
A cone is formed from a circle of
radius 10 cm with a 60° sector
removed. Another cone is formed
from a circle of radius 15 cm with a
60° sector removed. How do the two
cones differ? How are they the same?
The slant height of a cone is doubled. Does this double the surface
area of the cone? Explain your reasoning.
• Practise
For help with questions I and 2, see the Example.
1. Calculate the surface area of each cone. Round to the nearest
square unit.
c)
b)
a)
Am
8.4 cm
/\o cm
-
3.7 cm
10cm
2. a) Find the slant height of the cone,
b) Calculate the surface area of the
cone. Round to the nearest
square metre.
///
1am\
Connect and Apply
3. Some paper cups are shaped like cones.
a) I-low much paper, to the nearest square
centimetre, is needed to make the cup?
b) What assumptions have you made?
—Scm
i-.
12cm
8.4 Surface Area of a Cone• MHR 447
R,a,onIng and Provinc
RL’prnsenrn
(
S&ecilng TooIn
PrbIL’t71
.flectthg
ComnunIcatInt
-
4. One cone has base radius 4 cm and height 6 cm. Another cone has
a base radius 6 cm and height 4 cm.
a) Do the cones have the same slant height?
b) Do the cones have the same surface area? If not, predict which
cone has the greater surface area. Explain your reasoning.
c) Determine the surface area of each cone to check your prediction.
Were you correct?
5. The lateral area of a cone with radius 4 cm is 60 cm2
a) Determine the sLant height of the cone, to the nearest centimetre
b) Determine the height of the cone, to the nearest centimetre.
6. The height of a cone is doubled. Does this double the surface area?
Justih your answer.
7. The radius of a cone is doubled. Does this double the surface area?
Justify your answer.
8. A cube-shaped box has sides 10 cm in length.
a) What are the dimensions of the largest cone that fits inside this box?
b) What is the surface area of this cone, to the nearest square
centimetre?
9. A cone just fits inside a cylinder. The volume of the cylinder is
9425 cm3. What is the surface area of this cone, to the nearest square
centimetre?
-7”
20cm
10. The frustum of a cone is the part that remains after the top portion
has been removed by making a cut parallel to the base. Calculate the
surface area of this frustum, to the nearest square metre,
F
m
448 MHR ‘Chapters
iT
B
cIr.Dter Problem Emily has obtained an unfinished ceramic birdbath
for one of her customers. She plans to paint it with a special glaze so
that it will he weatherproof. The birdbath is constructed of two parts:
a shallow open-topped cylinder with an outside diameter of 1 m
and a depth of 5 cm, with I-cm-thick walls and base
a ( onical frustum on which the G3t1 inder sits
,‘-
20 cm
10cm
-H
60cm
1
m
a) Identify the surfaces that are to be painted and describe how
to calculate the area
b) CaLculate the surface area to be painted, to the nearest square
centimetre.
c) One can of glaze covers 1 in2. How many cans of glaze will Emily
need to cover all surfaces of the birdbath and the frustum?
12. Create a problem involving the surface area of a cone, Solve
the problem. Exchange with a classmate.
Extend
13. Suppose the cube in question B has sides of length x
a) Writeexpressions for the dimensions of the largest
cone that (its inside this box.
b) What is a formula for the surface area of this cone?
14. a) Find an expression for the slant height of a cone in terms
of its lateral area and its radius,
b) If the lateral area of a cone is 100 cm2 and its radius is
4 cm, determine its slant height.
15. Located in the Azores Islands off the coast of Portugal, Mt. Pico
Volcano stands 2351 m tall. Measure the photo to estimate the radius
of the base of the volcano, and then calculate its lateral surface area,
to the nearest square metre
rn
Did You Know?
There are 8000 to 10 000
people of Azorean heritage
living in Ontario.
8.4 surface Area of a Cone
‘
MHR 449
Communicate Your Understanding
O A cylindrical container and a conical container have the same
radius and height. How are their volumes related? I-low could
you illustrate this relationship for a friend?
O Suppose the height of a cone is doubled, Flow will this affect
the volume?
O Suppose the radius of a cone is doubled. How will this affect
the volume?
Practise
For help with question 1, see Example I
1. Determine the volume of each cone. Round to the nearest cubic unit.
b)
a)
/1\
/‘
cm
./\
\
c)
12
mm
m
/1
2cm
d)
cm
‘-‘30mm
40cm
For help with questions 2 and 3. soc Example 2.
2. Determine the volume of each cone. Round to the nearest cubic: unit.
b)
a)
//\
/
/ \\3Ocm
1m
10cm
3. Wesley uses a cone-shaped funnel to put oil in
a car engine. The funnel has a radius of 5.4 cm
and a slant height of 10.2 cm. How much oil can
the Funnel hold, to the nearest tenth of a cubic
10,2
centimetre?
454 MHR ‘Chapter 8
54cm
For help
iii
ii question 4, see Example 3.
of 67 cm3 and a diameter
4. A cone-shaped paper cup has a volume
t tenth
of 6 cm. What is the height of the paper cup, to the neares
of a centimetre?
Connect and Apply
volume 300 cm3
5. A cone just fits inside a cvi inder with
What is the volume of the cone?
I
a cone. Solve it.
5. Create a problem involving the volume of
Exchange your irohiem with a classmate.
volume of a
7. A cone has a volume of 150 cm3. What is the
cylinder that just holds the cone?
ce
8. A cone-shaped storage unit at a highway maintenan
s
depot holds 4000 m3 of sand. The unit has a base radiu
of 15 m,
a) Estimate the height of the storage unit.
b) Calculate the height.
c) How close was your estimate?
A
Li z
s of 3 cm.
9. A cone has a height of 4 cm and a base radiu
s of 4 cm.
Another cone has a height of 3 cm and a base radiu
Explain your
a) Predict which cone has the greater volume.
prediction.
nearest cubic
b) Calculate We volume of each cone, to the
?
correct
n
centimetre. Was your predictio
Determine
10. Chapter Problem Refer to question 11 in Section 8.4.
Round your answer
the volume of concrete in Emily’s birdbath.
to the nearest cubic centimetre.
EO cm
10cm
1cm
60 cm
Im
terms of its volume and its radius.
11. a) Express the height of a cone in
what is its height?
b) If a cone holds I L and its radius is 4 cm,
etre.
Round your answer to the nearest tenth of a centim
of water. If the height of the
12. A cone-shaped funnel holds 120 mL
ded to the nearest
s,
funnel is 15 cm, determine the radiu roun
tenth of a centimetre.
8.5 Volume ola ConeS MHR 455
Extend
13. A cone just fits inside a cube with sides that measure 10 cm.
a) What are the dimensions of the largest cone
that fits inside this box?
b) Estimate the ratio of the volume of the cone
to the volume of the cube.
c) Calculate the volume of the cone! to the
nearest cubic centimetre.
d) Calculate the ratio in part b).
e) How close was your estimate?
a height equal to its diameter. If the volume of the
m, determine the height of the cone, to the nearest
tenth of a metre.
14. A cone has
cone is 200
15. Use Technology Use a graphing calculator, The Ocoineter’s
Sketchpad®. or a spreadsheet to investigate how the volume of a
cone is affected when its rad ilis is constant and its height changes.
16. Use Technology A cone has a height of 20 cm.
a) Write an algebraic model for the volume of the cone in terms
of the radius.
b) Choose a toul for graphing. Graph the volume of the cone versus
the radius.
c) Describe the relationship using mathematical terms
17. Math Contest A cube has side length 6 cm. A square-based pvranii’
has side length 0 cm and height 12 cm. A cone has diameter 0 cm
and height 32 cm, A cylinder has diameter 6 cm and height 6 cm.
Order the figures from the least to the greatest volume. Select the
correct order
A cube! pyramid, cone, cylinder
B cylinder, cube, cone, pyramid
C cube, cone, cylinder, pyramid
D cone, pyramid, cylinder, cube
E pyramid, cone, cylinder, cube
456 MHR
ChapterS
Communicate Your Understanding
O
unt of leather
Describe how you would determine the amo
required to cover a softball.
the surface
Does doubling the radius of a sphere double
area? Explain your reasoning.
• Practise
ple I
For help with questions I and 2, see Exam
re. Round to the
1. Determine the surface area o each sphe
nearest square unit.
b)
a)
-
30.2 mm
6cm
d)
c)
3m
5.6
m
eter of 40 mm.
2. A ball used to play table tennis has a diam
a) Estimate the surface area of this ball
est square
b) Calculate the surface area, to the near
ate?
millimetre. How close was your estim
f//i
tN
Far help with question 3, see Example 2.
Find its radius,
3. A sphere has a surface area of 42,5 in2.
to the nearest tenth of a metre.
Connect and Apply
cm.
4. A basketball has a diameter of 24.8
r this ball,
a) How much leather is required to cove
metre?
to the nearest tenth of a square centi
does it cost to
b) If the loather costs $28/rn2, what
cover the basketball?
oximately 12 800 km.
S. The diameter of Earth is appr
, to the nearest square kilometre.
a) Calculate the surface area of Earth
e?
b) What assumptions did you mak
8.6 Surface Area of a SphereS MHR 459
p
6. a) The diameter of Mars is 6800 km. Calculate its
surface area, to the nearest square kilometre.
b) Compare the surface area of Mars to the surface
area of Earth from question 5. Approximately
how many times greater is the surface area of
Earth than the surface area of Mars?
in one
7. Chapter Problem Emily is placing a gazing ball
of her customer’s gardens. The ball has a diameter
s.
of 60 cm and will be covered with reflective crystal
m2.
One jar of these crystals covers 1
a) Estimate the surface area to decide whether one
jar of the crystals will cover the ball.
centimetre.
b) Calculate the surface area, to the nearest square
c) Was your estimate reasonable? Explain.
B. The radius of a sphere is 15 cm.
a) Predict how much the surface area increases if the radius
increases by 2 cm.
nearest
b) Calculate the change in the surface area, to the
square centimetre.
c) lion’ accurate was your prediction?
9. Use Technology
a sphere
a) Use a graphing calculator to graph the surface area of
versus its radius by entering the surface area formula.
b) Describe the relationship.
c) Use the TRACE feature to determine
• the surface area of a sphere with radius 5.35 cm
• the radius of a sphere with surface area 80 cm2
Extend
10. Use Technology
a) Determine an algebraic expression for the radius of a sphere in
terms of its surface area.
to
b) Use your expression from part a) and a graphing calculator
graph the relationship between the radius and the surface area.
c) Describe the relationship.
d) Use the graphing calculator to find the radius of a sphere with
surface area 200 cm2.
a
11. A spherical balloon is blown up from a diameter of 10 cm to
ed?
diameter of 30 cm. By what factor has its surface area increas
Explain your reasoning.
cube
12. Which has the Larger surface area: a sphere of radius r or a
with edges of length 2r?
460 MHR Chapter 8
ing
communicate Your Understand
o
r
ne the volume of a sphere
Describe how you would determi
if you knew its surface area.
if you double
How is the volume of a sphere affected
the radius?
Practise
see Example 1.
For lie/p with questions ito 3,
unit.
ere- Round to the nearest cubic
1. Calculate the volume of each sph
c)
b)
a)
(32mm
2.1 m
cm. Calculate its
2. A golf ball has a diameter of 4.3
etre.
tim
volume, to the nearest cubic cen
rrr
of baseballs killed
3. Hailstones thought to be the size
the Moradabad and Beheri
hundreds of people and cattle in
stones had a reported
districts of India in 1888. The haii
volume of each one, to the
diameter of 8 cm, What was the
nearest cubic centimetre?
r
mple 2.
For lie/p with question 4, see Exa
40 mm.
inside a plastic cube with edges
4. A table tennis ball just fits
table tennis ball, to the nearest
a) Calculate the volume of the
cubic millimetre.
cube.
b) Calculate the volume of the
ty space
c) Determine the amount of emp
8.7 Volume of a Sphere. MHR 465
Connect and Apply
S. The largest lollipop ever made had a diameter of 140,3 cm and
was made for a festival in Gränna, Swoden, on July 27, 2003.
a) If a spherical lollipop with diameter 4 cm has a mass of 50 g,
what was the mass of this giant lollipop to the nearest kilogram?
b) Describe any assumptions you have made.
6. Chapter Problem Emily orders a spherical gazing ball for one of
her customers. It is packaged tightly in a cylindrical container
with a base radius of 30 cm and a height of 60 cm.
a) Calculate the volume of the sphere, to the nearest cubic
centimetre.
b) Calculate the volume of the cylindrical container, to the nearest
cubic centimetre.
c) What is the ratio of the volume of the sphere to the volume
of the container?
d) Is this ratio consistent for any sphere that just fits inside
a cylinder? Explain your reasoning.
7. Golf balls are stacked three high in a rectangular prism
package. The diameter of one ball is 4.3 cm. What is the
minimum amount of material needed to make the box?
8.
A cylindrical silo has a hemispherical top (half of a
sphere). The cylinder has a height of 20 m and a base
diameter of 6.5 m.
a) Estimate the total volume of the silo.
b) Calculate the total volume, to the nearest cubic metre.
c) The silo should be filled to no more than 80% capacity
to allow for air circulation. 1-low much grain can be put
in the silo?
d) A truck with a bin measuring 7 m by 3 m by 2.5 m delivers grain
to the farm. How many truckloads would fill the silo to its
recommended capacity?
9. The tank of a propane tank truck
is in the shape of a cylinder with
a hemisphere at both ends. The
tank has a radius of 2 m and a total
length of 10.2 m. Calculate the
volume of the tank, to the nearest
cubic metre.
466 MHR’ Chapters
.
etballs would fit into your classroom
10. Estimate how many bask
estimation lechniques and describe any
Explain your reasoning and
your answer with that of a
assumptions you have made. Compare
whose answer is a more
classmate. Are your answers close? If not,
reasonable estimate and why?
k
• Achievement Chec
idering packaging two tennis balls that
1. The T-Ball company is cons
in a square-based prism.
are 8.5 cm in diameter in a cylinder or
•
of the two containers?
a) What are the dimensions and volumes
be in each container?
b) How much empty space would there
consider in choosing
c) What factors should Ge T-Ball company
the package design? Justify your choices.
Extend
1
.
a sphere with a volume of
Estimate and then calculate the radius of
600 cm3.
13. Use Technology Graph V
=
vr using a graphing calculator.
mine the volume of a sphere with
a) Use the TRACE feature to deter
a radius of 6.2 cm.
to
12 by using the TRACE feature
b) Check your answer to question
.
cm3
600
of
of a sphere with a volume
approximate the radius
is doubled from 250 cm2 to 500 cm2,
14. If the surface area of a sphere
increase?
by what factor does its volume
sides of length 8 cm.
15. A sphere just fits inside a cube with
of the sphere to the volume of
a) Estimate the ratio of the volume
the cube.
re and the cube and their ratio.
b) Calculate tho volumes of the sphe
estimate?
How does your answer compare to your
c)
8 cm
8 cm
8 cm
a sphere of radius r or a cube with
16. \\ihicli has the larger volume:
edges of length 2r?
8.7 Volume of a Sphere• MHR 467
2
n’nfl.
I.
II
I
Chapter 8 Review
8.1 Apply the Pythagorean Theorem,
pages 418—425
1. Determine the perimeter and area of each
right triangle. Round answers to the nearest
tenth of a unit or square unit.
a)
8.2 cm
ins cm
b)
2. A 6-rn extension ladder leans against a
vertical wall with its base 2 m frorn the
wall. How high up the wall does the top
of the ladder reach? Round to the nearest
tenth of a metre.
8.2 Perimeter and Area of Composite Figures,
pages 426—43 5
3. Calculate the perimeter and area of each
figure. Round answers to the nearest tenth
of a unit or square unit, if necessary.
a)
4. The diagram shows a running track at a high
school. The track consists of two parallel
line segments, with a semicircle at each end.
The track is 10 rn wide.
10Dm
©
84m
64mD
a) Tyler runs on the inner edge of the track.
How far does he run in one lap, to the
nearest tenth of a metre?
b) Dylan runs on the outer edge. How far
does he run in one lap, to the neatest
tenth of a metre?
c) Find the difference
the distances
run by Tyler and Dylan.
between
8.3 Surface Area and Volume of Prisms and
Pyramids, pages 43 6—443
5. Calculate the surface area of each object.
Round answers to the nearest square unit,
if necessary.
a)
Sm
46m
3m________
10
9m
4cm
5 cm
b)
/
Scm
N
10
10cm
b) the Great Pyramid of Cheops, with a
height of about 147 rn and a base width
of about 230
rn
rr
%
ri!± -—4
470 MHR’ChapterS
1
F:
e of the tent.
6. a) Calculate the volum
150cm
310
280
cm
cm
uired to make
D) How much nylon is req
this tent?
you made
c) Describe any assumptions
in part b)
answer in part h)?
d) How reasonable is your
500 mL and has a
7. A cylindrical can holds
height of the
radius of 4 cm. Calculate the
timetre.
cen
a
can, to the nearest tenth of
pages 444—450
8.4 Surface Area of a Cone,
of a cone with a
8. Calculate the surface area
a height of 12 cm.
slant height of 13 cm and
centimetre.
Round to the nearest square
‘I’
12\crn,f’
1
ffic pylon has a
9. The cone portion of a tra
vertical height
diameter of 20 cm and a
surface area of the
of 35 cm. Calculate the
to the nearest
cone portion of the pylon,
that the bottom
square centimetre. ssume
of the cone is complete.
20cm
35 cm
451—456
8.5 Volume of a Cone, pages
10. A conical funnel holds
100 mL. If the height of
the finns! is 10 cm,
10cm
determine its radius, to
the nearest tenth of a
centimetre
cone that just ills
11. Calculate the volume of a
radius of 8 cm
inside a cylinder with a base
to the nearest
and a height of 10 cm. Round
volume of the
cubic centimetre. How does the
of the cylinder?
cone compare to the volume
e, pages 457—461
8.6 Surface Area of a Spher
meter of 21.8 cm.
12. A volleyball has a dia
ther required to
Calculate the amount of lea
nearest tenth of
cover the volleyball, to the
a square centime! re.
rth is about 12 800 km.
13. The diameter of Ea
Northern
a) Calcutate the area of the
t square
Hemisphere, to the neares
kilometre.
e you made?
b) What assumptions hav
ate
0 070 610 km2. Estim
C) Canada’s area is
Hemisphere
the fraction of the Northern
that Canada covers.
pages 462—469
8.7 Volume of a Sphere,
a soccer ball with
14. Calculate the volume of
the nearest tenth
a diameter of 22.3 cm, to
of a cubic centimetre.
stion 14 is packaged
15. The soccer ball in que
cube-shaped box.
so that it just fits inside a
of empty space
a) Estimate the amount
inside the box.
of empty space.
b) Calculate the amount
estimate?
c How close was your
ChapterS Review’ MHR 471
Multiple Choice
Short Response
For questions I to 5, select the best answer.
Show all steps to your solutions.
1. A sphere has a radius of 3 cm, lVhat is its
volume, to the nearest cubic centimetre?
A
B
C
D
[r
6. A candle is in the shape of a square-based
pyramid.
cm3
38cm3
113 cm3
85cm3
330
10cm
8 cm
2. What is the area of the figure, to the nearest
square centimetre?
10 cm
A 43 cm2
B 54cm2
/7cm
C 62cm2
D 73cm2
5cm
3.
A circular swimming pool has a diameter of
7.5 m. It is filled to a depth of 1.4 m. What
is the volume of water in the pool. to the
nearest litre?
A 61850L
B
14
7. A rectangular cardboard carton is designed
to hold six rolls of paper towel that are
28 cm high and 10 cm in dinmeter. Describe
how you would calculate the amount of
8.
m2
B 90Gm2
C 707m2
0 090m2
Compare the effects of doubling the radius
on the volume of a cylinder and a sphere.
Justify your answer with numerical
examples.
9. Calculate the surface area of the cone that
just fits inside a cylinder with a base radius
of 8 cm and a height of 10 cm. Round to the
nearest square centimetre.
10cm
5. What is the length of the unknown side
of the triangle, to the nearest tenth of
a millimetre?
A
a) How much wax is needed to create the
candle, to the nearest cubic centimetre?
b) How much plastic wrap, to the nearest
tenth of a square centimetre, would you
need to completely cover the candle?
What assumptions did you make?
cardboard required to make this carton.
4. A conical pile of road salt is 15 m high and
has a base diameter of 30 it. How much
plastic sheeting is required to cover the pile,
to the nearest square metre?
A 414
L
8cm
247400L
C 23501L
D 47124L
I.’
:
I
iii L I I
11,..,..JJA 1J
2.3mm
B 5.0mm
C 6.1mm
D 7.7 mm
8 cm
10. Determine the volume of a conical pile of
grain that is 10 m high with a base diameter
of 20 m. Round to the nearest cubic metre.
1Dm
4.2mm
2Dm
472 MHR ‘Chapter 8
Response
provide complete solutions,
ure 8.4cm in
i. Three tennis balls that meas
diametur arc stackud in a clindrica1 can
EXteLLded
a) Determine the minimum volume
of the can, to the nearest tenth of a
cubic centimetre.
b) Calculate the amount of aluminum
required to make the can, including
the top and bottom, Round to the
nearest square centimetre.
c) The can comes with a plastic lid to be
used once the can is opened. Find the
amount of plastic required for the lid.
Round to the nearest square centimetre.
dJ Describe any assumptions you have
made.
12. A rectangular carton holds 12 cylindrical
cans flat each contain three tennis balls
like the ones described in question 11.
f((
N’/(
OOuO
flflQb
a) How much empty space is in each can
of tennis balls, to the nearest tenth of a
cubic centimetre?
b) Draw a diagram to show the dimensions
of the carton.
c) I-low much empty space is in the carton
and cans once (he 12 cans are placed in
the carton?
d) What is tim minimum amount of
cardboard necessary to make this carton?
Chapter Problem Wrap-Up
one of Emily’s customers.
You are to design a fountain for the garden of
with a cone on top.
• The fountain will have a cylindrical base
of 1 m.
• The cylindrical base will have a diameter
rete.
conc
of
• The fountain is to be made
protective paint.
• The entire fountain is to be coated with
all
d) Concrete costs $100/rn3. Each litre of
a) Make a sketch of your design, showing
protective paint costs $17.50 and covers
dimensions.
5 m2. Find the total cost of the materials
b) How much concrete is needed to make the
needed to make the fountain.
fountain?
c) What is the surface area that needs to be
Chapter 8 Practice Test MHR 473
I
15. Answers will vary. Examples:
a) The five triangles funned by two adjacent sides of
PQRST (AABC, ABCO, and so on) are isesceles and
congruent WAS). So. all the acute anglcu in these
triangles are equal. Then, AABR, ABCS, ACUT.
ADEP, and AEAQ are all congruent (ASA). The
obtuse angles of these triangles are opposite to the
interior angles of PQRST. Thus, these nngles are all
equal. AUPT, AEPQ, AAQR, ABRS, and ACST are all
congruent WAS), so the sides of PQRST are all equal.
b) Yes; both ore regular pentagons.
*
c) By measuring the diagram
d) Ratio of areas is
17. a)
7.1.
b) 66
16. a)45
n(n
IABV
2.7
—
1)
b)
nb
—
b) 90°
6. a) 95°
c) c 145°, d 60°,c
85°,f= 95°
d)v=55°,w=50°,x=75°y=70°,z= 110°
7. Answers will vary. Examples:
a) The sum of the interior angles is 360°. Opposite
interior angles are equal. .djacert interior angles
are supplementac
b) The diagonals bisect each other and bisect the
area of the parallelogram.
8. Example: A LC = 9n°, LB 60°.LD = 120°
9. 2160°
10. 15
11. Answers will vary. Example: Run the fence along
the median from the right vertex of the lot.
12. a) hexagon
b) Yes, the sides aro equal, and measuring with a
protractor shows that the interior angles are equal.
c) 120°
3)
————
d) For regular polygons, the measure of the interior
angles increases as the number of sides increases.
-
—
Chapter 7 RevIew, pages 408—409
1. a) 110°
b) 125°
c) w = 75°, x = 105°, y = 135°, z = 30°
2. The exterior angle would be greater than 180°.
3. a) any obtuse triangle
b) impossible: third exterior angle would he
greater than 180°
c) any acute triangle
d) impossible; sum of exterior angle would
be less than 360°
4. a) 100°
b)b=105°,c= 70°,d05°,e100°,f=80’
52°. z = 128°
C) x = 52°. Y
5. a) Example: three 110° angles
b) impossible; sum of the interior angles weuld
he greater than 360°
c) Example: three 100° angles
d) impossible; sum of the exterior angles would
be greater than 360°
c) 1800°
b) 2080°
6. a) 720°
7.
8.
9.
I 0.
a) 108°
30
b) 140°
I.
2.
3.
4.
5.
C
B
B
0
B
574 MHR Answers
Get Ready, pages 414—417
1. a)9.6m
d) 13.2cm
2. a) 17,6cm
3.
4.
5.
6.
7.
0.
b) 26 cm
e) 90 m
b) 32.0 m
c) 6.3 mm
1)35mm
c) 219.9mm
d) 39.3 r:m
4Dni
b) 105.7 c&
a) 38.6 cm2
b) 60.45 cm2
a) 11.34 &
b) 2513 cm2
a) 52 m2
24 m3; 9425 cm1
a)
20 m
c) 157.5°
Answers t’il) vary.
DE connects the miçlpoints of AB and AC. Therefore,
the base and altitude of AADE are half those of AABC,
11. a) Each median divides the triangle into two triangles.
All of these triangles are congruent (SAS). The
medians are equal in lengUl since they are sides
of the congruent triangles.
b) False; any scalene triangle is a counter-example.
12.—i 3. Answers will vary.
Chapter? Test, pages 410—41
Chapter 8
6.5 m
c) 050 m1
b) 685 m2
9.—il. Answers will van’.
0.1 Apply the Pythagorean Theorem, pages 418—425
I.
2.
3.
4.
5.
6.
7.
8.
a) 10 cm
a) 15 cm
a) 24 cm2
a) 4.5 units
35 cm
38 m
hOrn
104.56 m
Li)
b)
b)
b)
13
9.2 m
34.1 ni2
2.8 units
c) 6.6 m
c) 7.7 rn
c) 5 units
d) 8.6 cm
d) 7.4 cm
9, 11 stones
10, 64 cm
11. 4Oft
2 4; 2
12. a) 2 2; 2 3;
24
23
22
2
2
2
spiral pattern, the
sild right triangles to the
2 Number of Triangles
—
—
by
e
-; ——
eas
—
incr
l
wil
area
of three
ci
ss
tlti
e
aus
bec
e
appropriat
13. a) This name is
rem.
theo
an
ore
the Pythag
who] e numbers satisfies
b) Yes.
e
orean triples. with som
c) Yes, they are Pythag
a.
and
in
of
es
restrictions on the valu
21
—-+
2
c) As you
Ii)
d) m>
a>
0
pages 420—435
a of Composite Figures,
8.2 Perimeter and Are
c)54m
b)2Ocm
1. a)52m
e) 24 cm
d) 52 cm
C) 30cm’
b) 104 m’
2. a) 370 mm’
f) 174 m2
0)322 cm’
d) 45cm2
’
32m
b)2
3. a)62m
c) To [inc1 the perimeter:
ne
orean theorem to determi
Slop 1: Use the Pythag
.
side
n
now
the length of the unk
ry
ions of the outer bounda
Step 2: Add the dimens
ter.
ime
to determine lie per
cIa
the formula for he area
To [ind tl,e area: Use
trapezoid.
4
b) 1 paint can C) $4.5
4, a) 1500 cm’
5. 300 nun’
6.—i. Answers will
8.
9.
10.
11.
12.
14.
vary.
a) 180 plants b) 48 m’
Answers will vary.
b) 36%
a) 1610 cm2
b)9m
a)2.2m
5400 cm’
a)SOm’
of
n is four times the area
b) The area of the law
one flower bed.
d
vertices of the inscribe
c) It is only true if the
nts.
square are at the midpoi
a,
quadruples the ama. Are
ius
rad
the
g
blin
Doo
15.
a,.
Are
X
4
a1
4,u2. So, Are
Area. = ,r(2r)’ or
144
80,
55,
34,
a)
16.
40.104,
b) areas: 1.2,6, 15,
vary.
c)—d) Answers will
17. 1:5
18. 40cm’
Pyramids,
Volume of Prisms and
8.3 Surface Area and
pages 436-443
b) 147 cm’
1. a) 279.65 cm’
b) 2 ma’
2. a) 2000 nlm’
b) 402 cm’
’
3. a) 700 mm
b) 10.35 In’
4. a) 400 cm’
b) 24 m’
5. a) 52 m’
6. 30 cm
b) 115 324 m’
7. a) 1 604 000 m’
8. 143.5 m
9. Yes
10. a) 30 cm
ies (humps/dimples) on
b) There are no irregularit
nf the juice container
the surface. Also, tile top
completely full.
is
er
tain
con
is flat and the
c) $202.30
b) 15 cans
11. a)47 m’
80 rn’
le:
mp
Exa
y.
var
l
12. a) Answers wil
L
000
92
b)
mm
c) 020 mm or 151i and 20
ume.
Example: Double the vol
y.
var
l
s
wil
wer
Ans
13. a)
cm’
480
m
new pris
b) original prism 240 cm’;
.
y.
Yes
var
l
wil
s
wer
Ans
c)
ht of a triangular prism
d) Yes; doubling the heig
the prism.
e
doubles the volum of
is three times the height
d
ami
pyr
the
15. The height of
of the prism.
Volume of pyramid
/n-h
Volume of prism
nf the
equal, then the height
are
es
umn
If the two vol
the prism
of
ht
heig
the
es
tim
s
pyramid must be tire
san,e for both.
because mc and I are the
50
b) 5105
16. a) 56 Sn’
SA = 2(1w + wh + ii)
17. a)
K 211) + (2/ K 2/mfl
21121 K 2w) + (2w
4w]s + 4/hi
= 214/il’
IC/I ÷ //ill
t
4(2(1w
e.
t times the old volum
b) The new volume is eigh
1w],
Volume,,,
K 2k
,,,V 2] K 2w
Vnlunie,,
=
Ouch
18. 48
e, pages 444-450
8.4 5urface Area of a Can
c) 141 cIa’
b) 1257 cm’
I. a) 0 m’
’
83m
b)2
2. a)13m
3. a)lSBcm’
er being
re is nu pap
b) Answers will vary. The
.
pod
ap
‘erl
o
4. a)Yes.
e
The slant height is the sam
b) No. The second cone.
e
con
ond
sec
the
ion rs.
for both, hot in the express
has the greater radius.
; yes
t) 141 cm’; 249 cm’
b)3c,n
5. a)Scm
ula for the
vary. Example: The form
6. No. Answers will
i,r’ + ,rm-s. When the
is SA
surface area of the cone
term ,vrs is changed. The
the
y
onl
bled
ht
height is dou
Hencu, doubling the heig
ed.
lter
term ,rr’ remains una
.
aren
ace
mle the surf
of a cone does net doul
mula for the
y. Example: The For
7. No. Answers will var
trr’ + ,rrs. When the
is SA
e
con
a
of
surface area
will quadruple a,, d the
,r’
term
radius is doubled, the
hce area
double. Hence, the surl
term ‘cr5 xvii I more than
inal
orig
the
bo mnre than double
of the new cone will
colic.
8. a) radius 5
b)254 f.m2
cm.
height 10 cm
9. i:toz cm’
10. lsnm’
Answers MHR 515
11. a) base of the frustum, lateral area of the frustum, top of
the frustum, outer walls of the cylinder, inner walls
of the cylinder, the thia strip of the cylinder, the
outer part of the base of the cylinder, the inner part
of the base of the cylinder
c) 4 cans
b) 34 382 cm2
12. Answers will vary.
13. a) radius
b) SA
14. a) s=
,
‘cx’
——‘
b) 5027 mm; Answers will vafl’.
3. 1.8 m
4. a) 1932.2 cm2
b) $5.41
x, slant height
=
-‘rX
b)s
7.96 cm
15. Answers will vary, about 72 000 000 m2
16. a)SA = 4’c + 2’cs
b) Graphs will van. Should be a set points along
a straight line.
c) Answers will vary. Example: ft is a linear relation.
83 Volume of a Cone, pages 451—456
b) 188 m3
1. a) 25 cm’
d) 25 133 cm’
c) 2827 cm’
b) 2964 cm’
2. a) 2 m’
3. 264.1 cm’
4. 7.1 cm
5. luocm’
6. Answers will vary.
7. 450cm!
8. a) Answers will van’. Example: 18 m
b) 16.98 m
c) Answer will vary.
9. a) Answers will vary. Example: The cone with base
radius of 4 cm has the greater volume. The formula
for the volume of a cone contains two factors of
and only one factor of h. Hence, the volume is more
dependent on r than on h.
b) Cone (height 4 cm. base radius 3 cm):
Volume = 38 cm!
Cone (height 3cm, base radius 4cm):
Volume
50cm’
10. 141 045 cm3
b)59.7cm
12. 2.8 cm
13. a) radius 5 cm, height 10 cm b) Estimates will vary.
1:4
c) 262 cm3
d) 1:3.82
e) Answers will vary.
14. 9.1 m
15. Answers will van’. Example: When the radius is
constant, a change in height produces a proportional
change in volume.
16. a) V
8.6 Surface Area of a Sphere, pages 457—461
25
Lateral Area
11. a) h=’f
17. 0
b) 11461mm2
1. a)452 cm2
d)99m’
c)28m2
2. a) Answers will vary, about 4800 mm-’
height
+
c) Answers will vary. Example: The relation is
increasing for all values of r greater than 0 (since
the radius cannot be negative). The growth rate is
non-linear.
5. a) 514 718 540 jun2
Assumption: Earth is a sphere
145 267 244 luia2
approximately 3.5 times greater
Answers will vary Example: 10800cm2. No; two
jars will be required.
b) 11 310 cm
c) Answers will vary. Example: Yes; whether von
b)
6. a)
b)
7. a)
use the approximate value or the exact va’ue, two
jars of reflective crystals are required to cover the
gazing ball.
8. a) Answers will van. Example: 750 cm2
b) 804 cm
c) Answers will vary.
9. a)
ywmsz
b) The radius must be greater than 0. As the radius
increases, the surface area also increases in a
non-linear pattern.
c) 360cm-’; 2.5cm
10. a) r
SA
9
4,7
b)
S
c) TIme radius and the surface area must be greater than
0. The trend between die two variables is non-linear
with the radius increasing as the surface area
increases but at a slower rate.
d) 4 cm
11. The surface area has increased hy a factor of nine.
4rr
4’c(3r)’
=
4’c(Or’)
=
12. TIme cube with edge length 2r.
=
13. a) Answers will vary. Example:
H
516 MHR Answer5
b) surface area of sphere
surface area of cube
c) Answers will vary.
d) 1:1.91
100’c;
600; ,7:6
=
I
8.7 volume of a Sphere, pages 462—469
1.
2.
3.
4.
5.
c) 5 m3
b) 137 258 mm
a) 11 994 cm3
42 cm’
268 cm3
c) 30 490 alm1
b) 64 000 mm1
a) 33 510 mm3
a) 70.16cm
p
b) Answers may vary. Example: The largest lollipo
as
the
etre
centim
cubic
per
hod the same mass
small lollipop.
c) 2:3
b) 169 646 cin
6. a) 113 097 cm3
d) Yes. When the sphere just flis inside the cylinder,
Ii = 2r. So,
4
Volume.1,.
9. 1158 cni
10. 3.1 cm
x Vnlume,;,.
11. 678 cmh Volume1
12. 1493.0 on2
13. a) 257 359 270 lan2
b) Earth is a sphere.
c) Answers will vary. Example: about 25
14. 5806.5 cm3
cm3
15. a) Answers will vary. Example: about 5200
b) 5283.07 cm
c) Answers will vary.
Practise Test, pages 472473
Volume,.U.1h.
=
4
3
1
2
2
3
7. 258.86 cm2
8. a) Answers will vary.
c) 588 i&
C
A
A
D
B
a) 213 cm3 of wax
being
b) 236.3 cm2; Assum p1 ion: No plastic cover is
overlapped.
7. Answers will vary. Example: 5080 cm— if the palier
two rolls
towels are stacked in three columns with
in each column.
8. Doubling the radius of a sphere will im rease
(lie volume eight times. Doubling the radius
1.
2.
3.
4.
5.
6.
3
b) 736 ni1
d) 12 tn,ckleads
9. 111 m3
10. Answers will vary.
12. Estimates will vary. Actual radius is 5.23 cm.
13. a) 098.3 cml
b) 5.2cm
14. by a factor of about 2.83
15. a) Estimates will var. Example: 1:2
268 cm;
b) Volume of the sphere
512 cni’; ,i:6
Volume of the cube
c) Answers will vary.
16. the cube
17. Answers will vary.
18. B
19. 365.88 cm3
of a cylinder will quadruple the volume.
9. 523 cm2
10. 1047 m3
c) 55 cm2
b) 776 cm1
11. a) 1396.5 cm:I
covers
d) Answers will var’. Example: The circular lid
the top of the cylitidrir al din with no side parts.
b)
1?. a) 465.5cm
25.2cm
25.2cm
33.6cm
c) 10 165.3
cm
d) 4657 cm
Review. pages 470—471
1. a) perimeter 32.0 cm; area 411 cm2
b) perimeter 28.4 un; area 31.2 cm2. 5,7m
3. a) perimeter 28 m; area 48 m2
b) perimeter 32.6 cm; area 61.8 cm1
c) 62.8 In
b) 463.9 m
4. a) 401.1 In
ni1
736
138
b)
5. a) 220 cm2
6. a) 6 510 080 cI,12
b) 256 (124 cm2
walls of
c) Answers lvii I vary. Example: The side
the tent are flat.
answer is fairly
d) Answers lviii vary. Example: The
you lv1uit the
tent,
a
g
electin
when
as
reasonable
as possihile.
side walls to be as flat and stretr lied
7. 9.9cm
8. 283 cm2
Chapter 9
Get Ready, pages 476—477
b) 38 nI: 76.56 m2
a) 60cm; 280 cm’
b) 3,8 cm, 1.1 cm
a) 25.1 cm; 50.3 0n
b) 114.39 nt3; 143.54 m2
304
cm2
a) 320 cm;
314 m3; 291 a,2
b)
a) 1847 cm3; 836 cm2
)307
2 cm 1; 1088 cm2
ii
rIm
1288
a) I) 3072 ccI;
b) Their volumes aro equal.
al.
c) The second container requires less materi
h) 251% cm3; 1084 rm2
6. a) I) 2513 cm’; 817 cm2
b) Their volumes are equal.
c) ‘l’he first container requires less material.
1.
2.
3.
4.
5.
Answers MHR 577