Find the volume of each prism.
1.
SOLUTION:
The volume V of a prism is V = Bh, where B is the
area of a base and histheheightoftheprism.
The volume is 108 cm3.
2.
SOLUTION:
The volume V of a prism is V = Bh, where B is the
area of a base and histheheightoftheprism.
3.the oblique rectangular prism shown.
SOLUTION:
If two solids have the same height h and the same
cross-sectional area B at every level, then they have
the same volume. So, the volume of a right prism and
an oblique one of the same height and cross sectional
area are same.
4.an oblique pentagonal prism with a base area of 42
square centimeters and a height of 5.2 centimeters
SOLUTION:
If two solids have the same height h and the same
cross-sectional area B at every level, then they have
the same volume. So, the volume of a right prism and
an oblique one of the same height and cross sectional
area are same.
Find the volume of each cylinder. Round to the
nearest tenth.
5.
SOLUTION:
6.
SOLUTION:
If two solids have the same height h and the same
cross-sectional area B at every level, then they have
the same volume. So, the volume of a right cylinder
and an oblique one of the same height and cross
sectional area are same.
7.a cylinder with a diameter of 16 centimeters and a
height of 5.1 centimeters
SOLUTION:
8.a cylinder with a radius of 4.2 inches and a height of
7.4 inches
SOLUTION:
9.MULTIPLE CHOICE A rectangular lap pool
measures 80 feet long by 20 feet wide. If it needs to
be filled to 4 feet deep and each cubic foot holds 7.5
gallons, how many gallons will it take to fill the lap
pool?
A 4000
B 6400
C 30,000
D 48,000
SOLUTION:
Each cubic foot holds 7.5 gallons of water. So, the
amount of water required to fill the pool is 6400(7.5)
=48,000.
Therefore, the correct choice is D.
CCSS SENSE-MAKINGFindthevolumeof
each prism.
10.
SOLUTION:
The base is a rectangle of length 3 in. and width 2 in.
Theheightoftheprismis5in.
11.
SOLUTION:
The base is a triangle with a base length of 11 m and
the corresponding height of 7 m. The height of the
prismis14m.
12.
SOLUTION:
The base is a right triangle with a leg length of 9 cm
and the hypotenuse of length 15 cm.
Use the Pythagorean Theorem to find the height of
the base.
Theheightoftheprismis6cm.
13.
SOLUTION:
If two solids have the same height h and the same
cross-sectional area B at every level, then they have
the same volume. So, the volume of a right prism and
an oblique one of the same height and cross sectional
area are same.
The volume V of a prism is V = Bh, where B is the
area of a base and histheheightoftheprism.
B = 11.4 ft2 and h = 5.1 ft. Therefore, the volume is
14.an oblique hexagonal prism with a height of 15
centimeters and with a base area of 136 square
centimeters
SOLUTION:
If two solids have the same height h and the same
cross-sectional area B at every level, then they have
the same volume. So, the volume of a right prism and
an oblique one of the same height and cross sectional
area are same.
15.a square prism with a base edge of 9.5 inches and a
height of 17 inches
SOLUTION:
If two solids have the same height h and the same
cross-sectional area B at every level, then they have
the same volume. So, the volume of a right prism and
an oblique one of the same height and cross sectional
area are same.
CCSS SENSE-MAKINGFind the volume of
each cylinder. Round to the nearest tenth.
16.
SOLUTION:
r = 5 yd and h = 18 yd
17.
SOLUTION:
r = 6 cm and h=3.6cm.
18.
SOLUTION:
r=5.5in.
Use the Pythagorean Theorem to find the height of
the cylinder.
Now you can find the volume.
19.
SOLUTION:
If two solids have the same height h and the same
cross-sectional area B at every level, then they have
the same volume. So, the volume of a right prism and
an oblique one of the same height and cross sectional
area are same.
r = 7.5 mm and h=15.2mm.
20.PLANTER A planter is in the shape of a
rectangular prism 18 inches long, inches deep,
and 12 inches high. What is the volume of potting soil
in the planter if the planter is filled to inches
below the top?
SOLUTION:
The planter is to be filled inches below the top, so
21.SHIPPING A box 18 centimeters by 9 centimeters
by 15 centimeters is being used to ship two
cylindrical candles. Each candle has a diameter of 9
centimeters and a height of 15 centimeters, as shown
at the right. What is the volume of the empty space
in the box?
SOLUTION:
The volume of the empty space is the difference of
volumes of the rectangular prism and the cylinders.
22.SANDCASTLES In a sandcastle competition,
contestants are allowed to use only water, shovels,
and 10 cubic feet of sand. To transport the correct
amount of sand, they want to create cylinders that
are 2 feet tall to hold enough sand for one contestant.
What should the diameter of the cylinders be?
SOLUTION:
V = 10 ft3 and h = 2 ft Use the formula to find r.
Therefore, the diameter of the cylinders should be
about 2.52 ft.
Find the volume of the solid formed by each
net.
23.
SOLUTION:
The middle piece of the net is the front of the solid.
The top and bottom pieces are the bases and the
pieces on the ends are the side faces. This is a
triangular prism.
One leg of the base 14 cm and the hypotenuse 31.4
cm. Use the Pythagorean Theorem to find the height
of the base.
Theheightoftheprismis20cm.
The volume V of a prism is V = Bh, where B is the
area of the base, histheheightoftheprism.
24.
SOLUTION:
The circular bases at the top and bottom of the net
indicate that this is a cylinder. If the middle piece
were a rectangle, then the prism would be right.
However, since the middle piece is a parallelogram, it
is oblique.
The radius is 1.8 m, the height is 4.8 m, and the slant
height is 6 m.
If two solids have the same height h and the same
cross-sectional area B at every level, then they have
the same volume. So, the volume of a right prism and
an oblique one of the same height and cross sectional
area are same.
25.FOOD A cylindrical can of baked potato chips has a
height of 27 centimeters and a radius of 4
centimeters. A new can is advertised as being 30%
larger than the regular can. If both cans have the
same radius, what is the height of the larger can?
SOLUTION:
The volume of the smaller can is
The volume of the new can is 130% of the smaller
can,withthesameradius.
The height of the new can will be 35.1 cm.
26.CHANGING DIMENSIONS A cylinder has a
radius of 5 centimeters and a height of 8 centimeters.
Describe how each change affects the volume of the
cylinder.
a. The height is tripled.
b. The radius is tripled.
c. Both the radius and the height are tripled.
d. The dimensions are exchanged.
SOLUTION:
a.
When the height is tripled, h = 3h.
When the height is tripled, the volume is multiplied by
3.
b. When the radius is tripled, r = 3r.
So, when the radius is tripled, the volume is multiplied
by 9.
c. When the height and the radius are tripled, r = 3r
and h = 3h.
Whentheheightandtheradiusaretripled,the
volume is multiplied by 27.
d. When the dimensions are exchanged, r = 8 and h
= 5 cm.
Compare to the original volume.
The volume is multiplied by .
27.SOIL A soil scientist wants to determine the bulk
density of a potting soil to assess how well a specific
plant will grow in it. The density
of the soil sample is the ratio of its weight to its
volume.
a. If the weight of the container with the soil is 20
pounds and the weight of the container alone is 5
pounds, what is the soil’s bulk density? b. Assuming
that all other factors are favorable, how well should a
plant grow in this soil if a bulk density of 0.018 pound
per square inch is desirable for root growth? Explain.
c. If a bag of this soil holds 2.5 cubic feet, what is its
weight in pounds?
SOLUTION:
a. First calculate the volume of soil in the pot. Then
divide the weight of the soil by the volume.
The weight of the soil is the weight of the pot with
soilminustheweightofthepot.
W = 20 – 5=15lbs.
The soil density is thus:
b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant
should grow fairly well.
c.
Find the volume of each composite solid. Round
to the nearest tenth if necessary.
28.
SOLUTION:
The solid is a combination of two rectangular prisms.
The base of one rectangular prism is 5 cm by 3 cm
and the height is 11 cm. The base of the other prism
is4cmby3cmandtheheightis5cm.
29.
SOLUTION:
The solid is a combination of a rectangular prism and
a right triangular prism. The total volume of the solid
is the sum of the volumes of the two rectangular
prisms.
30.
SOLUTION:
The solid is a combination of a rectangular prism and
twohalfcylinders.
31.MANUFACTURING A can 12 centimeters tall fits
into a rubberized cylindrical holder that is 11.5
centimeters tall, including 1 centimeter for the
thickness of the base of the holder. The thickness of
the rim of the holder is 1 centimeter. What is the
volume of the rubberized material that makes up the
holder?
SOLUTION:
The volume of the rubberized material is the
difference between the volumes of the container and
the space used for the can. The container has a
radius of andaheightof11.5cm.
The empty space used to keep the can has a radius
of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The
volume V of a cylinder is V = πr2h,wherer is the
radius of the base and histheheightofthecylinder.
Therefore, the volume of the rubberized material is
about 304.1 cm3.
Find each measure to the nearest tenth.
32.A cylindrical can has a volume of 363 cubic
centimeters. The diameter of the can is 9
centimeters. What is the height?
SOLUTION:
33.A cylinder has a surface area of 144πsquare inches
and a height of 6 inches. What is the volume?
SOLUTION:
Use the surface area formula to solve for r.
The radius is 6. Find the volume.
34.A rectangular prism has a surface area of 432
square inches, a height of 6 inches, and a width of 12
inches. What is the volume?
SOLUTION:
Use the surface area formula to find the length of the
base of the prism.
Find the volume.
35.ARCHITECTURE A cylindrical stainless steel
column is used to hide a ventilation system in a new
building. According to the specifications, the diameter
of the column can be between 30 centimeters and 95
centimeters. The height is to be 500 centimeters.
What is the difference in volume between the largest
and smallest possible column? Round to the nearest
tenth cubic centimeter.
SOLUTION:
The volume will be the highest when the diameter is
95 cm and will be the lowest when it is 30 cm.That is
whentheradiiare47.5cmand15cmrespectively.
Find the difference between the volumes.
36.CCSS MODELING The base of a rectangular
swimming pool is sloped so one end of the pool is 6
feet deep and the other end is 3 feet deep, as shown
in the figure. If the width is 15 feet, find the volume
of water it takes to fill the pool.
SOLUTION:
The swimming pool is a combination of a rectangular
prism and a trapezoidal prism. The base of the
rectangular prism is 6 ft by 10 ft and the height is 15
ft. The bases of the trapezoidal prism are 6 ft and 3
ft long and the height of the base is 10 ft. The height
of the trapezoidal prism is 15 ft. The total volume of
the solid is the sum of the volumes of the two
prisms.
37.CHANGING DIMENSIONS A soy milk company
is planning a promotion in which the volume of soy
milk in each container will be increased by 25%. The
company wants the base of the container to stay the
same. What will be the height of the new containers?
SOLUTION:
Find the volume of the original container.
The volume of the new container is 125% of the
original container, with the same base dimensions.
Use 1.25V and B to find h.
38.DESIGN Sketch and label (in inches) three different
designs for a dry ingredient measuring cup that holds
1 cup. Be sure to include the dimensions in each
drawing.(1cup≈14.4375in3)
SOLUTION:
Sample answers:
For any cylindrical container, we have the following
equation for volume:
The last equation gives us a relation between the
radius and height of the cylinder that must be fulfilled
to get the desired volume. First, choose a suitable
radius,say1.85in,andsolvefortheheight.
If we choose a height of say 4 in., then we can solve
for the radius.
For any rectangular container, the volume equation is:
Choose numbers for any two of the dimensions and
we can solve for the third. Let l = 2.25 in. and w =
2.5in.
39.Find the volume of the regular pentagonal prism by
dividing it into five equal triangular prisms. Describe
the base area and height of each triangular prism.
SOLUTION:
The base of the prism can be divided into 5
congruent triangles of a base 8 cm and the
corresponding height 5.5 cm. So, the pentagonal
prism is a combination of 5 triangular prisms of height
10 cm. Find the base area of each triangular prism.
Therefore, the volume of the pentagonal prism is
40.PATIOS Mr. Thomas is planning to remove an old
patio and install a new rectangular concrete patio 20
feet long, 12 feet wide, and 4 inches thick. One
contractor bid $2225 for the project. A second
contractor bid $500 per cubic yard for the new patio
and $700 for removal of the old patio. Which is the
less expensive option? Explain.
SOLUTION:
Convert all of the dimensions to yards.
20 feet = yd
12 feet = 4 yd
4 in. = yd
Find the volume.
The total cost for the second contractor is about
.
Therefore, the second contractor is a less expensive
option.
41.MULTIPLE REPRESENTATIONS In this
problem, you will investigate right and oblique
cylinders
.
a. GEOMETRIC Draw a right cylinder and an
oblique cylinder with a height of 10 meters and a
diameter of 6 meters.
b. VERBAL A square prism has a height of 10
meters and a base edge of 6 meters. Is its volume
greater than, less than, or equal to the volume of
the cylinder? Explain.
c. ANALYTICAL Describe which change affects
the volume of the cylinder more: multiplying the
height by x or multiplying the radius by x. Explain.
SOLUTION:
a. The oblique cylinder should look like the right
cylinder (same height and size), except that it is
pushed a little to the side, like a slinky.
b. Find the volume of each.
The volume of the square prism is greater.
c. Do each scenario.
Assuming x > 1, multiplying the radius by x makes
the volume x2 times greater.
For example, if x = 0.5, then x2 = 0.25, which is less
than x.
42.CCSS CRITIQUE Franciso and Valerie each
calculated the volume of an equilateral triangular
prism with an apothem of 4 units and height of 5
units. Is either of them correct? Explain your
reasoning.
SOLUTION:
Francisco; Valerie incorrectly used asthe
lengthofonesideofthetriangularbase.Francisco
used a different approach, but his solution is correct.
Francisco used the standard formula for the volume
of a solid, V = Bh. The area of the base, B, is one-
half the apothem multiplied by the perimeter of the
base.
43.CHALLENGE A cylindrical can is used to fill a
container with liquid. It takes three full cans to fill the
container. Describe possible dimensions of the
container if it is each of the following shapes.
a. rectangular prism
b. square prism
c. triangular prism with a right triangle as the base
SOLUTION:
The volume of the can is 20πin3. It takes three full
cans to fill the container, so the volume of the
container is 60π in3.
a. Choose some basic values for 2 of the sides, and
then determine the third side. Base: 3 by 5.
3 by 5 by 4π
b. Choose some basic values for 2 of the sides, and
then determine the third side. Base: 5 by 5.
5 by 5 by
c. Choose some basic values for 2 of the sides, and
then determine the third side. Base: Legs: 3 by 4.
3 by 4 by 10π
44.WRITING IN MATH Write a helpful response to
the following question posted on an Internet
gardening forum.
I am new to gardening. The nursery will deliver a
truckload of soil, which they say is 4 yards. I
know that a yard is 3 feet, but what is a yard of
soil? How do I know what to order?
SOLUTION:
Sample answer: The nursery means a cubic yard,
which is 33 or 27 cubic feet. Find the volume of your
garden in cubic feet and divide by 27 to determine
the number of cubic yards of soil needed.
45.OPEN ENDED Draw and label a prism that has a
volume of 50 cubic centimeters.
SOLUTION:
Choose 3 values that multiply to make 50. The
factors of 50 are 2, 5, 5, so these are the simplest
values to choose.
Sample answer:
46.REASONING Determine whether the following
statement is true or false. Explain.
Two cylinders with the same height and the same
lateral area must have the same volume.
SOLUTION:
True; if two cylinders have the same height (h1 = h2)
and the same lateral area (L1 = L2), the circular
basesmusthavethesamearea.
The radii must also be equal.
47.WRITING IN MATH How are the formulas for
the volume of a prism and the volume of a cylinder
similar? How are they different?
SOLUTION:
Both formulas involve multiplying the area of the
base by the height. The base of a prism is a polygon,
so the expression representing the area varies,
depending on the type of polygon it is. The base of a
cylinder is a circle, so its area is πr2.
48.The volume of a triangular prism is 1380 cubic
centimeters. Its base is a right triangle with legs
measuring 8 centimeters and 15 centimeters. What is
the height of the prism?
A 34.5 cm
B 23 cm
C 17 cm
D 11.5 cm
SOLUTION:
49.A cylindrical tank used for oil storage has a height
that is half the length of its radius. If the volume of
the tank is 1,122,360 ft3, what is the tank’s radius?
F 89.4 ft
G 178.8 ft
H 280.9 ft
J 561.8 ft
SOLUTION:
50.SHORT RESPONSE What is the ratio of the area
of the circle to the area of the square?
SOLUTION:
The radius of the circle is 2x and the length of each
side of the square is 4x. So, the ratio of the areas can
be written as shown.
51.SAT/ACT A county proposes to enact a new 0.5%
property tax. What would be the additional tax
amount for a landowner whose property has a
taxable value of $85,000?
A $4.25
B $170
C $425
D $4250
E $42,500
SOLUTION:
Find the 0.5% of $85,000.
Therefore, the correct choice is C.
Find the lateral area and surface area of each
regular pyramid. Round to the nearest tenth if
necessary.
52.
SOLUTION:
The lateral area L of a regular pyramid is ,
where istheslantheightandP is the perimeter of
the base.
The slant height is the height of each of the
congruent lateral triangular faces. Use the
Pythagorean Theorem to find the slant height.
Find the perimeter and area of the equilateral
triangle for the base. Use the Pythagorean Theorem
to find the height h of the triangle.
The perimeter is P=3×10or30feet.
So, the area of the base B is ft2.
Find the lateral area L and surface area S of the
regular pyramid.
So, the lateral area of the pyramid is about 212.1 ft2.
Therefore, the surface area of the pyramid is about
255.4 ft2.
53.
SOLUTION:
The lateral area L of a regular pyramid is ,
where istheslantheightandP is the perimeter of
the base.
Here, the base is a square of side 7 cm and the slant
height is 9 cm.
So, the lateral area of the pyramid is 126 cm2.
The surface area S of a regular
pyramid is , where
L is the lateral area and B is the area of the base.
Therefore, the surface area of the pyramid is 175
cm2.
54.
SOLUTION:
The pyramid has a slant height of 15 inches and the
base is a hexagon with sides of 10.5 inches.
A central angle of the hexagon is or60°,sothe
angleformedinthetrianglebelowis30°.
Use a trigonometric ratio to find the measure of the
apothem a.
Find the lateral area and surface area of the
pyramid.
So, the lateral area of the pyramid is 472.5 in2.
Therefore, the surface area of the pyramid is about
758.9 in2.
55.BAKING Many baking pans are given a special
nonstick coating. A rectangular cake pan is 9 inches
by 13 inches by 2 inches deep. What is the area of
the inside of the pan that needs to be coated?
SOLUTION:
The area that needs to be coated is the sum of the
lateralareaandonebasearea.
Therefore, the area that needs to be coated is 2(13 +
9)(2) + 13(9) = 205 in2.
Find the indicated measure. Round to the
nearest tenth.
56.The area of a circle is 54 square meters. Find the
diameter.
SOLUTION:
The diameter of the circle is about 8.3 m.
57.Find the diameter of a circle with an area of 102
square centimeters.
SOLUTION:
The diameter of the circle is about 11.4 m.
58.The area of a circle is 191 square feet. Find the
radius.
SOLUTION:
59.Find the radius of a circle with an area of 271 square
inches.
SOLUTION:
Find the area of each trapezoid, rhombus, or
kite.
60.
SOLUTION:
The area A of a kite is one half the product of the
lengths of its diagonals, d1 and d2.
d1 = 12 in. and d2=7+13=20in.
61.
SOLUTION:
The area A of a trapezoid is one half the product of
the height h and the sum of the lengths of its bases,
b1 and b2.
62.
SOLUTION:
d1 = 2(22) = 44 and d2=2(23)=46
Find the volume of each prism.
1.
SOLUTION:
The volume V of a prism is V = Bh, where B is the
area of a base and histheheightoftheprism.
The volume is 108 cm3.
2.
SOLUTION:
The volume V of a prism is V = Bh, where B is the
area of a base and histheheightoftheprism.
3.the oblique rectangular prism shown.
SOLUTION:
If two solids have the same height h and the same
cross-sectional area B at every level, then they have
the same volume. So, the volume of a right prism and
an oblique one of the same height and cross sectional
area are same.
4.an oblique pentagonal prism with a base area of 42
square centimeters and a height of 5.2 centimeters
SOLUTION:
If two solids have the same height h and the same
cross-sectional area B at every level, then they have
the same volume. So, the volume of a right prism and
an oblique one of the same height and cross sectional
area are same.
Find the volume of each cylinder. Round to the
nearest tenth.
5.
SOLUTION:
6.
SOLUTION:
If two solids have the same height h and the same
cross-sectional area B at every level, then they have
the same volume. So, the volume of a right cylinder
and an oblique one of the same height and cross
sectional area are same.
7.a cylinder with a diameter of 16 centimeters and a
height of 5.1 centimeters
SOLUTION:
8.a cylinder with a radius of 4.2 inches and a height of
7.4 inches
SOLUTION:
9.MULTIPLE CHOICE A rectangular lap pool
measures 80 feet long by 20 feet wide. If it needs to
be filled to 4 feet deep and each cubic foot holds 7.5
gallons, how many gallons will it take to fill the lap
pool?
A 4000
B 6400
C 30,000
D 48,000
SOLUTION:
Each cubic foot holds 7.5 gallons of water. So, the
amount of water required to fill the pool is 6400(7.5)
=48,000.
Therefore, the correct choice is D.
CCSS SENSE-MAKINGFindthevolumeof
each prism.
10.
SOLUTION:
The base is a rectangle of length 3 in. and width 2 in.
Theheightoftheprismis5in.
11.
SOLUTION:
The base is a triangle with a base length of 11 m and
the corresponding height of 7 m. The height of the
prismis14m.
12.
SOLUTION:
The base is a right triangle with a leg length of 9 cm
and the hypotenuse of length 15 cm.
Use the Pythagorean Theorem to find the height of
the base.
Theheightoftheprismis6cm.
13.
SOLUTION:
If two solids have the same height h and the same
cross-sectional area B at every level, then they have
the same volume. So, the volume of a right prism and
an oblique one of the same height and cross sectional
area are same.
The volume V of a prism is V = Bh, where B is the
area of a base and histheheightoftheprism.
B = 11.4 ft2 and h = 5.1 ft. Therefore, the volume is
14.an oblique hexagonal prism with a height of 15
centimeters and with a base area of 136 square
centimeters
SOLUTION:
If two solids have the same height h and the same
cross-sectional area B at every level, then they have
the same volume. So, the volume of a right prism and
an oblique one of the same height and cross sectional
area are same.
15.a square prism with a base edge of 9.5 inches and a
height of 17 inches
SOLUTION:
If two solids have the same height h and the same
cross-sectional area B at every level, then they have
the same volume. So, the volume of a right prism and
an oblique one of the same height and cross sectional
area are same.
CCSS SENSE-MAKINGFind the volume of
each cylinder. Round to the nearest tenth.
16.
SOLUTION:
r = 5 yd and h = 18 yd
17.
SOLUTION:
r = 6 cm and h=3.6cm.
18.
SOLUTION:
r=5.5in.
Use the Pythagorean Theorem to find the height of
the cylinder.
Now you can find the volume.
19.
SOLUTION:
If two solids have the same height h and the same
cross-sectional area B at every level, then they have
the same volume. So, the volume of a right prism and
an oblique one of the same height and cross sectional
area are same.
r = 7.5 mm and h=15.2mm.
20.PLANTER A planter is in the shape of a
rectangular prism 18 inches long, inches deep,
and 12 inches high. What is the volume of potting soil
in the planter if the planter is filled to inches
below the top?
SOLUTION:
The planter is to be filled inches below the top, so
21.SHIPPING A box 18 centimeters by 9 centimeters
by 15 centimeters is being used to ship two
cylindrical candles. Each candle has a diameter of 9
centimeters and a height of 15 centimeters, as shown
at the right. What is the volume of the empty space
in the box?
SOLUTION:
The volume of the empty space is the difference of
volumes of the rectangular prism and the cylinders.
22.SANDCASTLES In a sandcastle competition,
contestants are allowed to use only water, shovels,
and 10 cubic feet of sand. To transport the correct
amount of sand, they want to create cylinders that
are 2 feet tall to hold enough sand for one contestant.
What should the diameter of the cylinders be?
SOLUTION:
V = 10 ft3 and h = 2 ft Use the formula to find r.
Therefore, the diameter of the cylinders should be
about 2.52 ft.
Find the volume of the solid formed by each
net.
23.
SOLUTION:
The middle piece of the net is the front of the solid.
The top and bottom pieces are the bases and the
pieces on the ends are the side faces. This is a
triangular prism.
One leg of the base 14 cm and the hypotenuse 31.4
cm. Use the Pythagorean Theorem to find the height
of the base.
Theheightoftheprismis20cm.
The volume V of a prism is V = Bh, where B is the
area of the base, histheheightoftheprism.
24.
SOLUTION:
The circular bases at the top and bottom of the net
indicate that this is a cylinder. If the middle piece
were a rectangle, then the prism would be right.
However, since the middle piece is a parallelogram, it
is oblique.
The radius is 1.8 m, the height is 4.8 m, and the slant
height is 6 m.
If two solids have the same height h and the same
cross-sectional area B at every level, then they have
the same volume. So, the volume of a right prism and
an oblique one of the same height and cross sectional
area are same.
25.FOOD A cylindrical can of baked potato chips has a
height of 27 centimeters and a radius of 4
centimeters. A new can is advertised as being 30%
larger than the regular can. If both cans have the
same radius, what is the height of the larger can?
SOLUTION:
The volume of the smaller can is
The volume of the new can is 130% of the smaller
can,withthesameradius.
The height of the new can will be 35.1 cm.
26.CHANGING DIMENSIONS A cylinder has a
radius of 5 centimeters and a height of 8 centimeters.
Describe how each change affects the volume of the
cylinder.
a. The height is tripled.
b. The radius is tripled.
c. Both the radius and the height are tripled.
d. The dimensions are exchanged.
SOLUTION:
a.
When the height is tripled, h = 3h.
When the height is tripled, the volume is multiplied by
3.
b. When the radius is tripled, r = 3r.
So, when the radius is tripled, the volume is multiplied
by 9.
c. When the height and the radius are tripled, r = 3r
and h = 3h.
Whentheheightandtheradiusaretripled,the
volume is multiplied by 27.
d. When the dimensions are exchanged, r = 8 and h
= 5 cm.
Compare to the original volume.
The volume is multiplied by .
27.SOIL A soil scientist wants to determine the bulk
density of a potting soil to assess how well a specific
plant will grow in it. The density
of the soil sample is the ratio of its weight to its
volume.
a. If the weight of the container with the soil is 20
pounds and the weight of the container alone is 5
pounds, what is the soil’s bulk density? b. Assuming
that all other factors are favorable, how well should a
plant grow in this soil if a bulk density of 0.018 pound
per square inch is desirable for root growth? Explain.
c. If a bag of this soil holds 2.5 cubic feet, what is its
weight in pounds?
SOLUTION:
a. First calculate the volume of soil in the pot. Then
divide the weight of the soil by the volume.
The weight of the soil is the weight of the pot with
soilminustheweightofthepot.
W = 20 – 5=15lbs.
The soil density is thus:
b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant
should grow fairly well.
c.
Find the volume of each composite solid. Round
to the nearest tenth if necessary.
28.
SOLUTION:
The solid is a combination of two rectangular prisms.
The base of one rectangular prism is 5 cm by 3 cm
and the height is 11 cm. The base of the other prism
is4cmby3cmandtheheightis5cm.
29.
SOLUTION:
The solid is a combination of a rectangular prism and
a right triangular prism. The total volume of the solid
is the sum of the volumes of the two rectangular
prisms.
30.
SOLUTION:
The solid is a combination of a rectangular prism and
twohalfcylinders.
31.MANUFACTURING A can 12 centimeters tall fits
into a rubberized cylindrical holder that is 11.5
centimeters tall, including 1 centimeter for the
thickness of the base of the holder. The thickness of
the rim of the holder is 1 centimeter. What is the
volume of the rubberized material that makes up the
holder?
SOLUTION:
The volume of the rubberized material is the
difference between the volumes of the container and
the space used for the can. The container has a
radius of andaheightof11.5cm.
The empty space used to keep the can has a radius
of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The
volume V of a cylinder is V = πr2h,wherer is the
radius of the base and histheheightofthecylinder.
Therefore, the volume of the rubberized material is
about 304.1 cm3.
Find each measure to the nearest tenth.
32.A cylindrical can has a volume of 363 cubic
centimeters. The diameter of the can is 9
centimeters. What is the height?
SOLUTION:
33.A cylinder has a surface area of 144πsquare inches
and a height of 6 inches. What is the volume?
SOLUTION:
Use the surface area formula to solve for r.
The radius is 6. Find the volume.
34.A rectangular prism has a surface area of 432
square inches, a height of 6 inches, and a width of 12
inches. What is the volume?
SOLUTION:
Use the surface area formula to find the length of the
base of the prism.
Find the volume.
35.ARCHITECTURE A cylindrical stainless steel
column is used to hide a ventilation system in a new
building. According to the specifications, the diameter
of the column can be between 30 centimeters and 95
centimeters. The height is to be 500 centimeters.
What is the difference in volume between the largest
and smallest possible column? Round to the nearest
tenth cubic centimeter.
SOLUTION:
The volume will be the highest when the diameter is
95 cm and will be the lowest when it is 30 cm.That is
whentheradiiare47.5cmand15cmrespectively.
Find the difference between the volumes.
36.CCSS MODELING The base of a rectangular
swimming pool is sloped so one end of the pool is 6
feet deep and the other end is 3 feet deep, as shown
in the figure. If the width is 15 feet, find the volume
of water it takes to fill the pool.
SOLUTION:
The swimming pool is a combination of a rectangular
prism and a trapezoidal prism. The base of the
rectangular prism is 6 ft by 10 ft and the height is 15
ft. The bases of the trapezoidal prism are 6 ft and 3
ft long and the height of the base is 10 ft. The height
of the trapezoidal prism is 15 ft. The total volume of
the solid is the sum of the volumes of the two
prisms.
37.CHANGING DIMENSIONS A soy milk company
is planning a promotion in which the volume of soy
milk in each container will be increased by 25%. The
company wants the base of the container to stay the
same. What will be the height of the new containers?
SOLUTION:
Find the volume of the original container.
The volume of the new container is 125% of the
original container, with the same base dimensions.
Use 1.25V and B to find h.
38.DESIGN Sketch and label (in inches) three different
designs for a dry ingredient measuring cup that holds
1 cup. Be sure to include the dimensions in each
drawing.(1cup≈14.4375in3)
SOLUTION:
Sample answers:
For any cylindrical container, we have the following
equation for volume:
The last equation gives us a relation between the
radius and height of the cylinder that must be fulfilled
to get the desired volume. First, choose a suitable
radius,say1.85in,andsolvefortheheight.
If we choose a height of say 4 in., then we can solve
for the radius.
For any rectangular container, the volume equation is:
Choose numbers for any two of the dimensions and
we can solve for the third. Let l = 2.25 in. and w =
2.5in.
39.Find the volume of the regular pentagonal prism by
dividing it into five equal triangular prisms. Describe
the base area and height of each triangular prism.
SOLUTION:
The base of the prism can be divided into 5
congruent triangles of a base 8 cm and the
corresponding height 5.5 cm. So, the pentagonal
prism is a combination of 5 triangular prisms of height
10 cm. Find the base area of each triangular prism.
Therefore, the volume of the pentagonal prism is
40.PATIOS Mr. Thomas is planning to remove an old
patio and install a new rectangular concrete patio 20
feet long, 12 feet wide, and 4 inches thick. One
contractor bid $2225 for the project. A second
contractor bid $500 per cubic yard for the new patio
and $700 for removal of the old patio. Which is the
less expensive option? Explain.
SOLUTION:
Convert all of the dimensions to yards.
20 feet = yd
12 feet = 4 yd
4 in. = yd
Find the volume.
The total cost for the second contractor is about
.
Therefore, the second contractor is a less expensive
option.
41.MULTIPLE REPRESENTATIONS In this
problem, you will investigate right and oblique
cylinders
.
a. GEOMETRIC Draw a right cylinder and an
oblique cylinder with a height of 10 meters and a
diameter of 6 meters.
b. VERBAL A square prism has a height of 10
meters and a base edge of 6 meters. Is its volume
greater than, less than, or equal to the volume of
the cylinder? Explain.
c. ANALYTICAL Describe which change affects
the volume of the cylinder more: multiplying the
height by x or multiplying the radius by x. Explain.
SOLUTION:
a. The oblique cylinder should look like the right
cylinder (same height and size), except that it is
pushed a little to the side, like a slinky.
b. Find the volume of each.
The volume of the square prism is greater.
c. Do each scenario.
Assuming x > 1, multiplying the radius by x makes
the volume x2 times greater.
For example, if x = 0.5, then x2 = 0.25, which is less
than x.
42.CCSS CRITIQUE Franciso and Valerie each
calculated the volume of an equilateral triangular
prism with an apothem of 4 units and height of 5
units. Is either of them correct? Explain your
reasoning.
SOLUTION:
Francisco; Valerie incorrectly used asthe
lengthofonesideofthetriangularbase.Francisco
used a different approach, but his solution is correct.
Francisco used the standard formula for the volume
of a solid, V = Bh. The area of the base, B, is one-
half the apothem multiplied by the perimeter of the
base.
43.CHALLENGE A cylindrical can is used to fill a
container with liquid. It takes three full cans to fill the
container. Describe possible dimensions of the
container if it is each of the following shapes.
a. rectangular prism
b. square prism
c. triangular prism with a right triangle as the base
SOLUTION:
The volume of the can is 20πin3. It takes three full
cans to fill the container, so the volume of the
container is 60π in3.
a. Choose some basic values for 2 of the sides, and
then determine the third side. Base: 3 by 5.
3 by 5 by 4π
b. Choose some basic values for 2 of the sides, and
then determine the third side. Base: 5 by 5.
5 by 5 by
c. Choose some basic values for 2 of the sides, and
then determine the third side. Base: Legs: 3 by 4.
3 by 4 by 10π
44.WRITING IN MATH Write a helpful response to
the following question posted on an Internet
gardening forum.
I am new to gardening. The nursery will deliver a
truckload of soil, which they say is 4 yards. I
know that a yard is 3 feet, but what is a yard of
soil? How do I know what to order?
SOLUTION:
Sample answer: The nursery means a cubic yard,
which is 33 or 27 cubic feet. Find the volume of your
garden in cubic feet and divide by 27 to determine
the number of cubic yards of soil needed.
45.OPEN ENDED Draw and label a prism that has a
volume of 50 cubic centimeters.
SOLUTION:
Choose 3 values that multiply to make 50. The
factors of 50 are 2, 5, 5, so these are the simplest
values to choose.
Sample answer:
46.REASONING Determine whether the following
statement is true or false. Explain.
Two cylinders with the same height and the same
lateral area must have the same volume.
SOLUTION:
True; if two cylinders have the same height (h1 = h2)
and the same lateral area (L1 = L2), the circular
basesmusthavethesamearea.
The radii must also be equal.
47.WRITING IN MATH How are the formulas for
the volume of a prism and the volume of a cylinder
similar? How are they different?
SOLUTION:
Both formulas involve multiplying the area of the
base by the height. The base of a prism is a polygon,
so the expression representing the area varies,
depending on the type of polygon it is. The base of a
cylinder is a circle, so its area is πr2.
48.The volume of a triangular prism is 1380 cubic
centimeters. Its base is a right triangle with legs
measuring 8 centimeters and 15 centimeters. What is
the height of the prism?
A 34.5 cm
B 23 cm
C 17 cm
D 11.5 cm
SOLUTION:
49.A cylindrical tank used for oil storage has a height
that is half the length of its radius. If the volume of
the tank is 1,122,360 ft3, what is the tank’s radius?
F 89.4 ft
G 178.8 ft
H 280.9 ft
J 561.8 ft
SOLUTION:
50.SHORT RESPONSE What is the ratio of the area
of the circle to the area of the square?
SOLUTION:
The radius of the circle is 2x and the length of each
side of the square is 4x. So, the ratio of the areas can
be written as shown.
51.SAT/ACT A county proposes to enact a new 0.5%
property tax. What would be the additional tax
amount for a landowner whose property has a
taxable value of $85,000?
A $4.25
B $170
C $425
D $4250
E $42,500
SOLUTION:
Find the 0.5% of $85,000.
Therefore, the correct choice is C.
Find the lateral area and surface area of each
regular pyramid. Round to the nearest tenth if
necessary.
52.
SOLUTION:
The lateral area L of a regular pyramid is ,
where istheslantheightandP is the perimeter of
the base.
The slant height is the height of each of the
congruent lateral triangular faces. Use the
Pythagorean Theorem to find the slant height.
Find the perimeter and area of the equilateral
triangle for the base. Use the Pythagorean Theorem
to find the height h of the triangle.
The perimeter is P=3×10or30feet.
So, the area of the base B is ft2.
Find the lateral area L and surface area S of the
regular pyramid.
So, the lateral area of the pyramid is about 212.1 ft2.
Therefore, the surface area of the pyramid is about
255.4 ft2.
53.
SOLUTION:
The lateral area L of a regular pyramid is ,
where istheslantheightandP is the perimeter of
the base.
Here, the base is a square of side 7 cm and the slant
height is 9 cm.
So, the lateral area of the pyramid is 126 cm2.
The surface area S of a regular
pyramid is , where
L is the lateral area and B is the area of the base.
Therefore, the surface area of the pyramid is 175
cm2.
54.
SOLUTION:
The pyramid has a slant height of 15 inches and the
base is a hexagon with sides of 10.5 inches.
A central angle of the hexagon is or60°,sothe
angleformedinthetrianglebelowis30°.
Use a trigonometric ratio to find the measure of the
apothem a.
Find the lateral area and surface area of the
pyramid.
So, the lateral area of the pyramid is 472.5 in2.
Therefore, the surface area of the pyramid is about
758.9 in2.
55.BAKING Many baking pans are given a special
nonstick coating. A rectangular cake pan is 9 inches
by 13 inches by 2 inches deep. What is the area of
the inside of the pan that needs to be coated?
SOLUTION:
The area that needs to be coated is the sum of the
lateralareaandonebasearea.
Therefore, the area that needs to be coated is 2(13 +
9)(2) + 13(9) = 205 in2.
Find the indicated measure. Round to the
nearest tenth.
56.The area of a circle is 54 square meters. Find the
diameter.
SOLUTION:
The diameter of the circle is about 8.3 m.
57.Find the diameter of a circle with an area of 102
square centimeters.
SOLUTION:
The diameter of the circle is about 11.4 m.
58.The area of a circle is 191 square feet. Find the
radius.
SOLUTION:
59.Find the radius of a circle with an area of 271 square
inches.
SOLUTION:
Find the area of each trapezoid, rhombus, or
kite.
60.
SOLUTION:
The area A of a kite is one half the product of the
lengths of its diagonals, d1 and d2.
d1 = 12 in. and d2=7+13=20in.
61.
SOLUTION:
The area A of a trapezoid is one half the product of
the height h and the sum of the lengths of its bases,
b1 and b2.
62.
SOLUTION:
d1 = 2(22) = 44 and d2=2(23)=46
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12-4 Volumes of Prisms and Cylinders
