Find the volume of each pyramid.
1.
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
The base of this pyramid is a right triangle with legs
of 9 inches and 5 inches and the height of the
pyramid is 10 inches.
2.
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
The base of this pyramid is a regular pentagon with
sides of 4.4 centimeters and an apothem of 3
centimeters. The height of the pyramid is 12
centimeters.
3.a rectangular pyramid with a height of 5.2 meters
and a base 8 meters by 4.5 meters
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
The base of this pyramid is a rectangle with a length
of 8 meters and a width of 4.5 meters. The height of
the pyramid is 5.2 meters.
4.a square pyramid with a height of 14 meters and a
base with 8-meter side lengths
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
The base of this pyramid is a square with sides of 8
meters. The height of the pyramid is 14 meters.
Find the volume of each cone. Round to the
nearest tenth.
5.
SOLUTION:
The volume of a circular cone is , or
, where B is the area of the base, h is the
height of the cone, and r is the radius of the base.
Since the diameter of this cone is 7 inches, the radius
is or 3.5 inches. The height of the cone is 4 inches.
6.
SOLUTION:
Use trigonometry to find the radius r.
The volume of a circular cone is , or
, where B is the area of the base, h is the
height of the cone, and r is the radius of the base.
The height of the cone is 11.5 centimeters.
7.an oblique cone with a height of 10.5 millimeters and
a radius of 1.6 millimeters
SOLUTION:
The volume of a circular cone is , or
, where B is the area of the base, h is the
height of the cone, and r is the radius of the base.
The radius of this cone is 1.6 millimeters and the
height is 10.5 millimeters.
8.a cone with a slant height of 25 meters and a radius
of 15 meters
SOLUTION:
Use the Pythagorean Theorem to find the height h of
the cone. Then find its volume.
So, the height of the cone is 20 meters.
9.MUSEUMS The sky dome of the National Corvette
Museum in Bowling Green, Kentucky, is a conical
building. If the height is 100 feet and the area of the
base is about 15,400 square feet, find the volume of
air that the heating and cooling systems would have
to accommodate. Round to the nearest tenth.
SOLUTION:
The volume of a circular cone is , where B
is the area of the base and h is the height of the
cone.
For this cone, the area of the base is 15,400 square
feet and the height is 100 feet.
CCSS SENSE-MAKINGFindthevolumeof
each pyramid. Round to the nearest tenth if
necessary.
10.
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
11.
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
12.
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
13.
SOLUTION:
The volume of a pyramid is , where B is the
base and histheheightofthepyramid.
The base is a hexagon, so we need to make a right tri
determine the apothem. The interior angles of the he
120°.Theradiusbisectstheangle,sotherighttriangl
90°triangle.
The apothem is .
14.a pentagonal pyramid with a base area of 590 square
feet and an altitude of 7 feet
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
15.a triangular pyramid with a height of 4.8 centimeters
and a right triangle base with a leg 5 centimeters and
hypotenuse 10.2 centimeters
SOLUTION:
Find the height of the right triangle.
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
16.A triangular pyramid with a right triangle base with a
leg 8 centimeters and hypotenuse 10 centimeters has
a volume of 144 cubic centimeters. Find the height.
SOLUTION:
The base of the pyramid is a right triangle with a leg
of 8 centimeters and a hypotenuse of 10 centimeters.
Use the Pythagorean Theorem to find the other leg a
of the right triangle and then find the area of the
triangle.
The length of the other leg of the right triangle is 6
cm.
So, the area of the base B is 24 cm2.
Replace V with 144 and B with 24 in the formula for
the volume of a pyramid and solve for the height h.
Therefore, the height of the triangular pyramid is 18
cm.
Find the volume of each cone. Round to the
nearest tenth.
17.
SOLUTION:
The volume of a circular cone is ,
wherer is the radius of the base and h is the height
of the cone.
Since the diameter of this cone is 10 inches, the
radius is or 5 inches. The height of the cone is 9
inches.
Therefore, the volume of the cone is about 235.6 in3.
18.
SOLUTION:
Thevolumeofacircularconeis , where
r is the radius of the base and h is the height of the
cone. The radius of this cone is 4.2 centimeters and
the height is 7.3 centimeters.
Therefore, the volume of the cone is about 134.8
cm3.
19.
SOLUTION:
Use a trigonometric ratio to find the height h of the
cone.
Thevolumeofacircularconeis , where
r is the radius of the base and h is the height of the
cone. The radius of this cone is 8 centimeters.
Therefore, the volume of the cone is about 1473.1
cm3.
20.
SOLUTION:
Use trigonometric ratios to find the height h and the
radius r of the cone.
Thevolumeofacircularconeis , where
r is the radius of the base and h is the height of the
cone.
Therefore, the volume of the cone is about 2.8 ft3.
21.an oblique cone with a diameter of 16 inches and an
altitude of 16 inches
SOLUTION:
The volume of a circular cone is , where
r is the radius of the base and h is the height of the
cone. Since the diameter of this cone is 16 inches,
the radius is or 8 inches.
Therefore, the volume of the cone is about 1072.3
in3.
22.a right cone with a slant height of 5.6 centimeters
and a radius of 1 centimeter
SOLUTION:
The cone has a radius r of 1 centimeter and a slant
height of 5.6 centimeters. Use the Pythagorean
Theorem to find the height h of the cone.
Therefore, the volume of the cone is about 5.8 cm3.
23.SNACKS Approximately how many cubic
centimeters of roasted peanuts will completely fill a
paper cone that is 14 centimeters high and has a base
diameter of 8 centimeters? Round to the nearest
tenth.
SOLUTION:
The volume of a circular cone is , where
r is the radius of the base and h is the height of the
cone. Since the diameter of the cone is 8
centimeters, the radius is or 4 centimeters. The
height of the cone is 14 centimeters.
Therefore, the paper cone will hold about 234.6 cm3
of roasted peanuts.
24.CCSS MODELING The Pyramid Arena in
Memphis, Tennessee, is the third largest pyramid in
the world. It is approximately 350 feet tall, and its
square base is 600 feet wide. Find the volume of this
pyramid.
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
25.GARDENING The greenhouse is a regular
octagonal pyramid with a height of 5 feet. The base
has side lengths of 2 feet. What is the volume of the
greenhouse?
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
The base of the pyramid is a regular octagon with
sides of 2 feet. A central angle of the octagon is
or45°,sotheangleformedinthetriangle
belowis22.5°.
Use a trigonometric ratio to find the apothem a.
The height of this pyramid is 5 feet.
Therefore, the volume of the greenhouse is about
32.2 ft3.
Find the volume of each solid. Round to the
nearest tenth.
26.
SOLUTION:
Volume of the solid given = Volume of the small
cone + Volume of the large cone
27.
SOLUTION:
28.
SOLUTION:
29.HEATING Sam is building an art studio in her
backyard. To buy a heating unit for the space, she
needs to determine the BTUs (British Thermal Units)
required to heat the building. For new construction
with good insulation, there should be 2 BTUs per
cubic foot. What size unit does Sam need to
purchase?
SOLUTION:
The building can be broken down into the rectangular
base and the pyramid ceiling. The volume of the base
is
The volume of the ceiling is
The total volume is therefore 5000 + 1666.67 =
6666.67 ft3. Two BTU's are needed for every cubic
foot, so the size of the heating unit Sam should buy is
6666.67×2=13,333BTUs.
30.SCIENCE Refer to page 825. Determine the
volume of the model. Explain why knowing the
volume is helpful in this situation.
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
It tells Marta how much clay is needed to make the
model.
31.CHANGING DIMENSIONS A cone has a radius
of 4 centimeters and a height of 9 centimeters.
Describe how each change affects the volume of the
cone.
a. The height is doubled.
b. The radius is doubled.
c. Both the radius and the height are doubled.
SOLUTION:
Find the volume of the original cone. Then alter the
values.
a. Double h.
The volume is doubled.
b. Double r.
The volume is multiplied by 22 or 4.
c. Double r and h.
volume is multiplied by 23 or 8.
Find each measure. Round to the nearest tenth
if necessary.
32.A square pyramid has a volume of 862.5 cubic
centimeters and a height of 11.5 centimeters. Find
the side length of the base.
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
Let s be the side length of the base.
The side length of the base is 15 cm.
33.The volume of a cone is 196πcubic inches and the
height is 12 inches. What is the diameter?
SOLUTION:
The volume of a circular cone is , or
, where B is the area of the base, h is the
height of the cone, and r is the radius of the base.
Since the diameter is 8 centimeters, the radius is 4
centimeters.
The diameter is 2(7) or 14 inches.
34.The lateral area of a cone is 71.6 square millimeters
and the slant height is 6 millimeters. What is the
volume of the cone?
SOLUTION:
The lateral area of a cone is , where r is
the radius and is the slant height of the cone.
Replace L with 71.6 and with 6, then solve for the
radius r.
So, the radius is about 3.8 millimeters.
Use the Pythagorean Theorem to find the height of
the cone.
So, the height of the cone is about 4.64 millimeters.
The volume of a circular cone is , where
r is the radius of the base and h is the height of the
cone.
Therefore, the volume of the cone is about 70.2
mm3.
35.MULTIPLE REPRESENTATIONS In this
problem, you will investigate rectangular pyramids.
a. GEOMETRIC Draw two pyramids with
different bases that have a height of 10 centimeters
and a base area of 24 square centimeters.
b. VERBAL What is true about the volumes of the
two pyramids that you drew? Explain.
c. ANALYTICAL Explain how multiplying the base
area and/or the height of the pyramid by 5 affects the
volume of the pyramid.
SOLUTION:
a. Use rectangular bases and pick values that
multiply to make 24.
Sample answer:
b. The volumes are the same. The volume of a
pyramid equals one third times the base area times
the height. So, if the base areas of two pyramids are
equal and their heights are equal, then their volumes
are equal.
c. If the base area is multiplied by 5, the volume is
multiplied by 5. If the height is multiplied by 5, the
volume is multiplied by 5. If both the base area and
the height are multiplied by 5, the volume is multiplied
by5·5or25.
36.CCSS ARGUMENTS Determine whether the
following statement is sometimes, always, or never
true. Justify your reasoning.
The volume of a cone with radius r and height h
equals the volume of a prism with height h.
SOLUTION:
The volume of a cone with a radius r and height h is
. The volume of a prism with a height of
h is where B is the area of the base of the
prism. Set the volumes equal.
The volumes will only be equal when the radius of
the cone is equal to or when .
Therefore, the statement is true sometimes if the
base area of the cone is 3 times as great as the base
area of the prism. For example, if the base of the
prism has an area of 10 square units, then its volume
is 10h cubic units. So, the cone must have a base
area of 30 square units so that its volume is
or10h cubic units.
37.ERROR ANALYSIS Alexandra and Cornelio are
calculating the volume of the cone below. Is either of
them correct? Explain your answer.
SOLUTION:
The slant height is used for surface area, but the
height is used for volume. For this cone, the slant
height of 13 is provided, and we need to calculate the
height before we can calculate the volume.
Alexandra incorrectly used the slant height.
38.REASONING A cone has a volume of 568 cubic
centimeters. What is the volume of a cylinder that
has the same radius and height as the cone? Explain
your reasoning.
SOLUTION:
1704 cm3; The formula for the volume of a cylinder
is V= Bh, while the formula for the volume of a cone
is V = Bh. The volume of a cylinder is three times
as much as the volume of a cone with the same
radius and height.
39.OPEN ENDED Give an example of a pyramid and
a prism that have the same base and the same
volume. Explain your reasoning.
SOLUTION:
The formula for volume of a prism is V = Bh and the
formula for the volume of a pyramid is one-third of
that. So, if a pyramid and prism have the same base,
then in order to have the same volume, the height of
the pyramid must be 3 times as great as the height of
the prism.
Set the base areas of the prism and pyramid, and
make the height of the pyramid equal to 3 times the
height of the prism.
Sample answer:
A square pyramid with a base area of 16 and a
height of 12, a prism with a square base area of 16
and a height of 4.
40.WRITING IN MATH Compare and contrast
finding volumes of pyramids and cones with finding
volumes of prisms and cylinders.
SOLUTION:
To find the volume of each solid, you must know the
area of the base and the height. The volume of a
pyramid is one third the volume of a prism that has
the same height and base area. The volume of a
cone is one third the volume of a cylinder that has the
same height and base area.
41.A conical sand toy has the dimensions as shown
below. How many cubic centimeters of sand will it
hold when it is filled to the top?
A 12π
B 15π
C
D
SOLUTION:
Use the Pythagorean Theorem to find the radius r of
the cone.
So, the radius of the cone is 3 centimeters.
Thevolumeofacircularconeis , where
r is the radius of the base and h is the height of the
cone.
Therefore, the correct choice is A.
42.SHORT RESPONSE Brooke is buying a tent that
is in the shape of a rectangular pyramid. The base is
6 feet by 8 feet. If the tent holds 88 cubic feet of air,
how tall is the tent’s center pole?
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
43.PROBABILITY A spinner has sections colored
red, blue, orange, and green. The table below shows
the results of several spins. What is the experimental
probability of the spinner landing on orange?
F
G
H
J
SOLUTION:
Possible outcomes: {6 red, 4 blue, 5 orange, 10
green}
Number of possible outcomes : 25
Favorable outcomes: {5 orange}
Number of favorable outcomes: 5
So, the correct choice is F.
44.SAT/ACT For all
A –8
B x – 4
C
D
E
SOLUTION:
So, the correct choice is E.
Find the volume of each prism.
45.
SOLUTION:
The volume of a prism is , where B is the
area of the base and h is the height of the prism. The
base of this prism is a rectangle with a length of 14
inches and a width of 12 inches. The height h of the
prism is 6 inches.
Therefore, the volume of the prism is 1008 in3.
46.
SOLUTION:
The volume of a prism is , where B is the
area of the base and h is the height of the prism. The
base of this prism is an isosceles triangle with a base
of 10 feet and two legs of 13 feet. The height h will
bisect the base. Use the Pythagorean Theorem to
find the height of the triangle.
So, the height of the triangle is 12 feet. Find the area
of the triangle.
So, the area of the base B is 60 ft2.
The height h of the prism is 19 feet.
Therefore, the volume of the prism is 1140 ft3.
47.
SOLUTION:
The volume of a prism is , where B is the
area of the base and h is the height of the prism. The
base of this prism is a rectangle with a length of 79.4
meters and a width of 52.5 meters. The height of the
prism is 102.3 meters.
Therefore, the volume of the prism is about 426,437.6
m3.
48.FARMING The picture shows a combination
hopper cone and bin used by farmers to store grain
after harvest. The cone at the bottom of the bin
allows the grain to be emptied more easily. Use the
dimensions in the diagram to find the entire surface
area of the bin with a conical top and bottom. Write
the exact answer and the answer rounded to the
nearest square foot.
Refer to the photo on Page 847.
SOLUTION:
To find the entire surface area of the bin, find the
surface area of the conical top and bottom and find
the surface area of the cylinder and add them.
The formula for finding the surface area of a cone is
, where is r is the radius and l is the slant height
of the cone.
Find the slant height of the conical top.
Find the slant height of the conical bottom.
The height of the conical bottom is 28 – (5 + 12 + 2)
or 9 ft.
The formula for finding the surface area of a cylinder
is , where is r is the radius and h is the slant
height of the cylinder.
Surface area of the bin = Surface area of the
cylinder + Surface area of the conical top + surface
area of the conical bottom.
Find the area of each shaded region. Polygons
in 50 - 52 are regular.
49.
SOLUTION:
Area of the shaded region = Area of the rectangle –
Area of the circle
Area of the rectangle = 10(5)
= 50
50.
SOLUTION:
Area of the shaded region = Area of the circle –
Area of the hexagon
A regular hexagon has 6 sides, so the measure of the
interior angle is . The apothem bisects the
angle,soweuse30°whenusingtrigtofindthe
lengthoftheapothem.
Now find the area of the hexagon.
51.
SOLUTION:
The shaded area is the area of the equilateral triangle
less the area of the inscribed circle.
Find the area of the circle with a radius of 3.6 feet.
So, the area of the circle is about 40.7 square feet.
Next, find the area of the equilateral triangle.
The equilateral triangle can be divided into three
isosceles triangles with central angles of or
120°,sotheangleinthetrianglecreatedbythe
heightoftheisoscelestriangleis60°andthebaseis
half the length b of the side of the equilateral triangle.
Useatrigonometricratiotofindthevalueofb.
Now use the area of the isosceles triangles to find
the area of the equilateral triangle.
So, the area of the equilateral triangle is about 67.3
square feet.
Subtract to find the area of shaded region.
Therefore, the area of the shaded region is about
26.6 ft2.
52.
SOLUTION:
The area of the shaded region is the area of the
circumscribed circle minus the area of the equilateral
triangle plus the area of the inscribed circle. Let b
represent the length of each side of the equilateral
triangle and h represent the radius of the inscribed
circle.
Divide the equilateral triangle into three isosceles
triangles with central angles of or120°.The
angle in the triangle formed by the height.h of the
isoscelestriangleis60°andthebaseishalfthe
length of the side of the equilateral triangle as shown
below.
Use trigonometric ratios to find the values of b and
h.
The area of the equilateral triangle will equal three
times the area of any of the isosceles triangles. Find
the area of the shaded region.
A(shaded region) = A(circumscribed circle) - A
(equilateral triangle) + A(inscribed circle)
Therefore, the area of the shaded region is about
168.2 mm2.
Find the volume of each pyramid.
1.
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
The base of this pyramid is a right triangle with legs
of 9 inches and 5 inches and the height of the
pyramid is 10 inches.
2.
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
The base of this pyramid is a regular pentagon with
sides of 4.4 centimeters and an apothem of 3
centimeters. The height of the pyramid is 12
centimeters.
3.a rectangular pyramid with a height of 5.2 meters
and a base 8 meters by 4.5 meters
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
The base of this pyramid is a rectangle with a length
of 8 meters and a width of 4.5 meters. The height of
the pyramid is 5.2 meters.
4.a square pyramid with a height of 14 meters and a
base with 8-meter side lengths
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
The base of this pyramid is a square with sides of 8
meters. The height of the pyramid is 14 meters.
Find the volume of each cone. Round to the
nearest tenth.
5.
SOLUTION:
The volume of a circular cone is , or
, where B is the area of the base, h is the
height of the cone, and r is the radius of the base.
Since the diameter of this cone is 7 inches, the radius
is or 3.5 inches. The height of the cone is 4 inches.
6.
SOLUTION:
Use trigonometry to find the radius r.
The volume of a circular cone is , or
, where B is the area of the base, h is the
height of the cone, and r is the radius of the base.
The height of the cone is 11.5 centimeters.
7.an oblique cone with a height of 10.5 millimeters and
a radius of 1.6 millimeters
SOLUTION:
The volume of a circular cone is , or
, where B is the area of the base, h is the
height of the cone, and r is the radius of the base.
The radius of this cone is 1.6 millimeters and the
height is 10.5 millimeters.
8.a cone with a slant height of 25 meters and a radius
of 15 meters
SOLUTION:
Use the Pythagorean Theorem to find the height h of
the cone. Then find its volume.
So, the height of the cone is 20 meters.
9.MUSEUMS The sky dome of the National Corvette
Museum in Bowling Green, Kentucky, is a conical
building. If the height is 100 feet and the area of the
base is about 15,400 square feet, find the volume of
air that the heating and cooling systems would have
to accommodate. Round to the nearest tenth.
SOLUTION:
The volume of a circular cone is , where B
is the area of the base and h is the height of the
cone.
For this cone, the area of the base is 15,400 square
feet and the height is 100 feet.
CCSS SENSE-MAKINGFindthevolumeof
each pyramid. Round to the nearest tenth if
necessary.
10.
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
11.
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
12.
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
13.
SOLUTION:
The volume of a pyramid is , where B is the
base and histheheightofthepyramid.
The base is a hexagon, so we need to make a right tri
determine the apothem. The interior angles of the he
120°.Theradiusbisectstheangle,sotherighttriangl
90°triangle.
The apothem is .
14.a pentagonal pyramid with a base area of 590 square
feet and an altitude of 7 feet
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
15.a triangular pyramid with a height of 4.8 centimeters
and a right triangle base with a leg 5 centimeters and
hypotenuse 10.2 centimeters
SOLUTION:
Find the height of the right triangle.
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
16.A triangular pyramid with a right triangle base with a
leg 8 centimeters and hypotenuse 10 centimeters has
a volume of 144 cubic centimeters. Find the height.
SOLUTION:
The base of the pyramid is a right triangle with a leg
of 8 centimeters and a hypotenuse of 10 centimeters.
Use the Pythagorean Theorem to find the other leg a
of the right triangle and then find the area of the
triangle.
The length of the other leg of the right triangle is 6
cm.
So, the area of the base B is 24 cm2.
Replace V with 144 and B with 24 in the formula for
the volume of a pyramid and solve for the height h.
Therefore, the height of the triangular pyramid is 18
cm.
Find the volume of each cone. Round to the
nearest tenth.
17.
SOLUTION:
The volume of a circular cone is ,
wherer is the radius of the base and h is the height
of the cone.
Since the diameter of this cone is 10 inches, the
radius is or 5 inches. The height of the cone is 9
inches.
Therefore, the volume of the cone is about 235.6 in3.
18.
SOLUTION:
Thevolumeofacircularconeis , where
r is the radius of the base and h is the height of the
cone. The radius of this cone is 4.2 centimeters and
the height is 7.3 centimeters.
Therefore, the volume of the cone is about 134.8
cm3.
19.
SOLUTION:
Use a trigonometric ratio to find the height h of the
cone.
Thevolumeofacircularconeis , where
r is the radius of the base and h is the height of the
cone. The radius of this cone is 8 centimeters.
Therefore, the volume of the cone is about 1473.1
cm3.
20.
SOLUTION:
Use trigonometric ratios to find the height h and the
radius r of the cone.
Thevolumeofacircularconeis , where
r is the radius of the base and h is the height of the
cone.
Therefore, the volume of the cone is about 2.8 ft3.
21.an oblique cone with a diameter of 16 inches and an
altitude of 16 inches
SOLUTION:
The volume of a circular cone is , where
r is the radius of the base and h is the height of the
cone. Since the diameter of this cone is 16 inches,
the radius is or 8 inches.
Therefore, the volume of the cone is about 1072.3
in3.
22.a right cone with a slant height of 5.6 centimeters
and a radius of 1 centimeter
SOLUTION:
The cone has a radius r of 1 centimeter and a slant
height of 5.6 centimeters. Use the Pythagorean
Theorem to find the height h of the cone.
Therefore, the volume of the cone is about 5.8 cm3.
23.SNACKS Approximately how many cubic
centimeters of roasted peanuts will completely fill a
paper cone that is 14 centimeters high and has a base
diameter of 8 centimeters? Round to the nearest
tenth.
SOLUTION:
The volume of a circular cone is , where
r is the radius of the base and h is the height of the
cone. Since the diameter of the cone is 8
centimeters, the radius is or 4 centimeters. The
height of the cone is 14 centimeters.
Therefore, the paper cone will hold about 234.6 cm3
of roasted peanuts.
24.CCSS MODELING The Pyramid Arena in
Memphis, Tennessee, is the third largest pyramid in
the world. It is approximately 350 feet tall, and its
square base is 600 feet wide. Find the volume of this
pyramid.
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
25.GARDENING The greenhouse is a regular
octagonal pyramid with a height of 5 feet. The base
has side lengths of 2 feet. What is the volume of the
greenhouse?
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
The base of the pyramid is a regular octagon with
sides of 2 feet. A central angle of the octagon is
or45°,sotheangleformedinthetriangle
belowis22.5°.
Use a trigonometric ratio to find the apothem a.
The height of this pyramid is 5 feet.
Therefore, the volume of the greenhouse is about
32.2 ft3.
Find the volume of each solid. Round to the
nearest tenth.
26.
SOLUTION:
Volume of the solid given = Volume of the small
cone + Volume of the large cone
27.
SOLUTION:
28.
SOLUTION:
29.HEATING Sam is building an art studio in her
backyard. To buy a heating unit for the space, she
needs to determine the BTUs (British Thermal Units)
required to heat the building. For new construction
with good insulation, there should be 2 BTUs per
cubic foot. What size unit does Sam need to
purchase?
SOLUTION:
The building can be broken down into the rectangular
base and the pyramid ceiling. The volume of the base
is
The volume of the ceiling is
The total volume is therefore 5000 + 1666.67 =
6666.67 ft3. Two BTU's are needed for every cubic
foot, so the size of the heating unit Sam should buy is
6666.67×2=13,333BTUs.
30.SCIENCE Refer to page 825. Determine the
volume of the model. Explain why knowing the
volume is helpful in this situation.
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
It tells Marta how much clay is needed to make the
model.
31.CHANGING DIMENSIONS A cone has a radius
of 4 centimeters and a height of 9 centimeters.
Describe how each change affects the volume of the
cone.
a. The height is doubled.
b. The radius is doubled.
c. Both the radius and the height are doubled.
SOLUTION:
Find the volume of the original cone. Then alter the
values.
a. Double h.
The volume is doubled.
b. Double r.
The volume is multiplied by 22 or 4.
c. Double r and h.
volume is multiplied by 23 or 8.
Find each measure. Round to the nearest tenth
if necessary.
32.A square pyramid has a volume of 862.5 cubic
centimeters and a height of 11.5 centimeters. Find
the side length of the base.
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
Let s be the side length of the base.
The side length of the base is 15 cm.
33.The volume of a cone is 196πcubic inches and the
height is 12 inches. What is the diameter?
SOLUTION:
The volume of a circular cone is , or
, where B is the area of the base, h is the
height of the cone, and r is the radius of the base.
Since the diameter is 8 centimeters, the radius is 4
centimeters.
The diameter is 2(7) or 14 inches.
34.The lateral area of a cone is 71.6 square millimeters
and the slant height is 6 millimeters. What is the
volume of the cone?
SOLUTION:
The lateral area of a cone is , where r is
the radius and is the slant height of the cone.
Replace L with 71.6 and with 6, then solve for the
radius r.
So, the radius is about 3.8 millimeters.
Use the Pythagorean Theorem to find the height of
the cone.
So, the height of the cone is about 4.64 millimeters.
The volume of a circular cone is , where
r is the radius of the base and h is the height of the
cone.
Therefore, the volume of the cone is about 70.2
mm3.
35.MULTIPLE REPRESENTATIONS In this
problem, you will investigate rectangular pyramids.
a. GEOMETRIC Draw two pyramids with
different bases that have a height of 10 centimeters
and a base area of 24 square centimeters.
b. VERBAL What is true about the volumes of the
two pyramids that you drew? Explain.
c. ANALYTICAL Explain how multiplying the base
area and/or the height of the pyramid by 5 affects the
volume of the pyramid.
SOLUTION:
a. Use rectangular bases and pick values that
multiply to make 24.
Sample answer:
b. The volumes are the same. The volume of a
pyramid equals one third times the base area times
the height. So, if the base areas of two pyramids are
equal and their heights are equal, then their volumes
are equal.
c. If the base area is multiplied by 5, the volume is
multiplied by 5. If the height is multiplied by 5, the
volume is multiplied by 5. If both the base area and
the height are multiplied by 5, the volume is multiplied
by5·5or25.
36.CCSS ARGUMENTS Determine whether the
following statement is sometimes, always, or never
true. Justify your reasoning.
The volume of a cone with radius r and height h
equals the volume of a prism with height h.
SOLUTION:
The volume of a cone with a radius r and height h is
. The volume of a prism with a height of
h is where B is the area of the base of the
prism. Set the volumes equal.
The volumes will only be equal when the radius of
the cone is equal to or when .
Therefore, the statement is true sometimes if the
base area of the cone is 3 times as great as the base
area of the prism. For example, if the base of the
prism has an area of 10 square units, then its volume
is 10h cubic units. So, the cone must have a base
area of 30 square units so that its volume is
or10h cubic units.
37.ERROR ANALYSIS Alexandra and Cornelio are
calculating the volume of the cone below. Is either of
them correct? Explain your answer.
SOLUTION:
The slant height is used for surface area, but the
height is used for volume. For this cone, the slant
height of 13 is provided, and we need to calculate the
height before we can calculate the volume.
Alexandra incorrectly used the slant height.
38.REASONING A cone has a volume of 568 cubic
centimeters. What is the volume of a cylinder that
has the same radius and height as the cone? Explain
your reasoning.
SOLUTION:
1704 cm3; The formula for the volume of a cylinder
is V= Bh, while the formula for the volume of a cone
is V = Bh. The volume of a cylinder is three times
as much as the volume of a cone with the same
radius and height.
39.OPEN ENDED Give an example of a pyramid and
a prism that have the same base and the same
volume. Explain your reasoning.
SOLUTION:
The formula for volume of a prism is V = Bh and the
formula for the volume of a pyramid is one-third of
that. So, if a pyramid and prism have the same base,
then in order to have the same volume, the height of
the pyramid must be 3 times as great as the height of
the prism.
Set the base areas of the prism and pyramid, and
make the height of the pyramid equal to 3 times the
height of the prism.
Sample answer:
A square pyramid with a base area of 16 and a
height of 12, a prism with a square base area of 16
and a height of 4.
40.WRITING IN MATH Compare and contrast
finding volumes of pyramids and cones with finding
volumes of prisms and cylinders.
SOLUTION:
To find the volume of each solid, you must know the
area of the base and the height. The volume of a
pyramid is one third the volume of a prism that has
the same height and base area. The volume of a
cone is one third the volume of a cylinder that has the
same height and base area.
41.A conical sand toy has the dimensions as shown
below. How many cubic centimeters of sand will it
hold when it is filled to the top?
A 12π
B 15π
C
D
SOLUTION:
Use the Pythagorean Theorem to find the radius r of
the cone.
So, the radius of the cone is 3 centimeters.
Thevolumeofacircularconeis , where
r is the radius of the base and h is the height of the
cone.
Therefore, the correct choice is A.
42.SHORT RESPONSE Brooke is buying a tent that
is in the shape of a rectangular pyramid. The base is
6 feet by 8 feet. If the tent holds 88 cubic feet of air,
how tall is the tent’s center pole?
SOLUTION:
The volume of a pyramid is , where B is the
area of the base and h is the height of the pyramid.
43.PROBABILITY A spinner has sections colored
red, blue, orange, and green. The table below shows
the results of several spins. What is the experimental
probability of the spinner landing on orange?
F
G
H
J
SOLUTION:
Possible outcomes: {6 red, 4 blue, 5 orange, 10
green}
Number of possible outcomes : 25
Favorable outcomes: {5 orange}
Number of favorable outcomes: 5
So, the correct choice is F.
44.SAT/ACT For all
A –8
B x – 4
C
D
E
SOLUTION:
So, the correct choice is E.
Find the volume of each prism.
45.
SOLUTION:
The volume of a prism is , where B is the
area of the base and h is the height of the prism. The
base of this prism is a rectangle with a length of 14
inches and a width of 12 inches. The height h of the
prism is 6 inches.
Therefore, the volume of the prism is 1008 in3.
46.
SOLUTION:
The volume of a prism is , where B is the
area of the base and h is the height of the prism. The
base of this prism is an isosceles triangle with a base
of 10 feet and two legs of 13 feet. The height h will
bisect the base. Use the Pythagorean Theorem to
find the height of the triangle.
So, the height of the triangle is 12 feet. Find the area
of the triangle.
So, the area of the base B is 60 ft2.
The height h of the prism is 19 feet.
Therefore, the volume of the prism is 1140 ft3.
47.
SOLUTION:
The volume of a prism is , where B is the
area of the base and h is the height of the prism. The
base of this prism is a rectangle with a length of 79.4
meters and a width of 52.5 meters. The height of the
prism is 102.3 meters.
Therefore, the volume of the prism is about 426,437.6
m3.
48.FARMING The picture shows a combination
hopper cone and bin used by farmers to store grain
after harvest. The cone at the bottom of the bin
allows the grain to be emptied more easily. Use the
dimensions in the diagram to find the entire surface
area of the bin with a conical top and bottom. Write
the exact answer and the answer rounded to the
nearest square foot.
Refer to the photo on Page 847.
SOLUTION:
To find the entire surface area of the bin, find the
surface area of the conical top and bottom and find
the surface area of the cylinder and add them.
The formula for finding the surface area of a cone is
, where is r is the radius and l is the slant height
of the cone.
Find the slant height of the conical top.
Find the slant height of the conical bottom.
The height of the conical bottom is 28 – (5 + 12 + 2)
or 9 ft.
The formula for finding the surface area of a cylinder
is , where is r is the radius and h is the slant
height of the cylinder.
Surface area of the bin = Surface area of the
cylinder + Surface area of the conical top + surface
area of the conical bottom.
Find the area of each shaded region. Polygons
in 50 - 52 are regular.
49.
SOLUTION:
Area of the shaded region = Area of the rectangle –
Area of the circle
Area of the rectangle = 10(5)
= 50
50.
SOLUTION:
Area of the shaded region = Area of the circle –
Area of the hexagon
A regular hexagon has 6 sides, so the measure of the
interior angle is . The apothem bisects the
angle,soweuse30°whenusingtrigtofindthe
lengthoftheapothem.
Now find the area of the hexagon.
51.
SOLUTION:
The shaded area is the area of the equilateral triangle
less the area of the inscribed circle.
Find the area of the circle with a radius of 3.6 feet.
So, the area of the circle is about 40.7 square feet.
Next, find the area of the equilateral triangle.
The equilateral triangle can be divided into three
isosceles triangles with central angles of or
120°,sotheangleinthetrianglecreatedbythe
heightoftheisoscelestriangleis60°andthebaseis
half the length b of the side of the equilateral triangle.
Useatrigonometricratiotofindthevalueofb.
Now use the area of the isosceles triangles to find
the area of the equilateral triangle.
So, the area of the equilateral triangle is about 67.3
square feet.
Subtract to find the area of shaded region.
Therefore, the area of the shaded region is about
26.6 ft2.
52.
SOLUTION:
The area of the shaded region is the area of the
circumscribed circle minus the area of the equilateral
triangle plus the area of the inscribed circle. Let b
represent the length of each side of the equilateral
triangle and h represent the radius of the inscribed
circle.
Divide the equilateral triangle into three isosceles
triangles with central angles of or120°.The
angle in the triangle formed by the height.h of the
isoscelestriangleis60°andthebaseishalfthe
length of the side of the equilateral triangle as shown
below.
Use trigonometric ratios to find the values of b and
h.
The area of the equilateral triangle will equal three
times the area of any of the isosceles triangles. Find
the area of the shaded region.
A(shaded region) = A(circumscribed circle) - A
(equilateral triangle) + A(inscribed circle)
Therefore, the area of the shaded region is about
168.2 mm2.
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12-5 Volumes of Pyramids and Cones
