olqghu 5rxqgwrwkh€¦ · )lqgwkhyroxphrihdfkf\olqghu 5rxqgwrwkh qhduhvwwhqwk 62/87,21 $16:(5 iw...

$: OLQGHU 5RXQGWRWKH€¦ · )lqgwkhyroxphrihdfkf\olqghu 5rxqgwrwkh qhduhvwwhqwk 62/87,21 $16:(5 iw 62/87,21 ,iwzrvrolgvkdyhwkhvdphkhljkw kdqgwkhvdph furvv vhfwlrqdoduhd %dwhyhu\ohyho$
Find the volume of each prism.

1.

SOLUTION:

The volume V of a prism is V = Bh, where B is thearea of a base and h is the height of the prism.

The volume is 108 cm3.

ANSWER:

108 cm3

2.

SOLUTION:

The volume V of a prism is V = Bh, where B is thearea of a base and h is the height of the prism.

ANSWER:

396 in3

3. the oblique rectangular prism shown

SOLUTION:

If two solids have the same height h and the samecross-sectional area B at every level, then they havethe same volume. So, the volume of a right prism andan oblique one of the same height and cross sectionalarea are same.

ANSWER:

26.95 m3

4. an oblique pentagonal prism with a base area of 42square centimeters and a height of 5.2 centimeters

SOLUTION:


ANSWER:

218.4 cm3

eSolutions Manual - Powered by Cognero Page 1

11-2 Volumes of Prisms and Cylinders

Find the volume of each cylinder. Round to thenearest tenth.

5.

SOLUTION:

ANSWER:

206.4 ft3

6.

SOLUTION:

If two solids have the same height h and the samecross-sectional area B at every level, then they havethe same volume. So, the volume of a right cylinderand an oblique one of the same height and crosssectional area are same.

ANSWER:

1357.2 m3

7. a cylinder with a diameter of 16 centimeters and aheight of 5.1 centimeters

SOLUTION:

ANSWER:

1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of7.4 inches

SOLUTION:

ANSWER:

410.1 in3



9. MULTIPLE CHOICE A rectangular lap poolmeasures 80 feet long by 20 feet wide. If it needs tobe filled to four feet deep and each cubic foot holds7.5 gallons, how many gallons will it take to fill the lappool?A 4000B 6400C 30,000D 48,000

SOLUTION:

Each cubic foot holds 7.5 gallons of water. So, theamount of water required to fill the pool is 6400(7.5)= 48,000. Therefore, the correct choice is D.

ANSWER:

D

STRUCTURE Find the volume of each prism.

10.

SOLUTION:

The base is a rectangle of length 3 in. and width 2 in.The height of the prism is 5 in.

ANSWER:

30 in3

11.

SOLUTION:

The base is a triangle with a base length of 11 m andthe corresponding height of 7 m. The height of theprism is 14 m.

ANSWER:

539 m3



12.

SOLUTION:

The base is a right triangle with a leg length of 9 cmand the hypotenuse of length 15 cm. Use the Pythagorean Theorem to find the height ofthe base.

The height of the prism is 6 cm.

ANSWER:

324 cm3

13.

SOLUTION:

If two solids have the same height h and the samecross-sectional area B at every level, then they havethe same volume. So, the volume of a right prism andan oblique one of the same height and cross sectionalarea are same. The volume V of a prism is V = Bh, where B is thearea of a base and h is the height of the prism.

B = 11.4 ft2 and h = 5.1 ft. Therefore, the volume is

ANSWER:

58.14 ft3

14. an oblique hexagonal prism with a height of 15centimeters and with a base area of 136 squarecentimeters

SOLUTION:


ANSWER:

2040 cm3



15. a square prism with a base edge of 9.5 inches and aheight of 17 inches

SOLUTION:


ANSWER:

1534.25 in3

STRUCTURE Find the volume of each cylinder.Round to the nearest tenth.

16.

SOLUTION:

r = 5 yd and h = 18 yd

ANSWER:

1413.7 yd3

17.

SOLUTION:

r = 6 cm and h = 3.6 cm.

ANSWER:

407.2 cm3

18.

SOLUTION:

r = 5.5 in. Use the Pythagorean Theorem to find the height ofthe cylinder.

Now you can find the volume.

ANSWER:

823.0 in3



19.

SOLUTION:

If two solids have the same height h and the samecross-sectional area B at every level, then they havethe same volume. So, the volume of a right prism andan oblique one of the same height and cross sectionalarea are same. r = 7.5 mm and h = 15.2 mm.

ANSWER:

2686.1 mm3

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

ANSWER:

2740.5 in3

21. SHIPPING The box shown is being used to shiptwo cylindrical candles. What is the volume of theempty space in the box?

SOLUTION:

The volume of the empty space is the difference ofvolumes of the rectangular prism and the cylinders.

The volume of the empty space is 521.5 cm³.

ANSWER:

521.5 cm3

22. CHANGING DIMENSIONS A cylinder has aradius of 5 centimeters and a height of 8 centimeters.Describe how each change affects the volume of thecylinder.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 multipliedby 9. c. When the height and the radius are tripled, r = 3rand h = 3h.

When the height and the radius are tripled, thevolume 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 .

ANSWER:

a. The volume is multiplied by 3.b. The volume is multiplied by 32 or 9.c. The volume is multiplied by 33 or 27.

d. The volume is multiplied by

23. INSULATION The insulated cup holds 16 ouncesof liquid. Find the volume of the insulating material,rounded to the nearest cubic inch.

SOLUTION:

The volume of the insulated material is the differencebetween the volumes of the interior cylinder (whichholds the liquid) and the entire cylinder (cup). Theinner cylinder has a volume of 16 ounces (whichconverts to 28.875 cubic inches). Use this to find theradius of the inner cylinder. Note that the height ofthe inner cylinder is , due to the extra0.5 insulation at the bottom of the cup.

Therefore, the radius of the inner cylinder is

inches, making the entire cup's radius to be

inches.

Find the volume of the entire cup.

The volume of the insulating material is the differencebetween the volume of the inner cylinder and thevolume of the entire cup.



Therefore, the volume of the insulated material isabout 31 in3.

ANSWER:

31 in3

24. MODELING The base of a rectangular paint tray issloped as shown below. Find the volume of paint ittakes to fill the tray.

SOLUTION:

The paint tray is a combination of a rectangular prismand a trapezoidal prism. The base of the rectangular prism is 8.9 cm by 45 cmand the height is 8.4 cm. The bases of the trapezoidal prism are 8.4 cm and 1.3cm and the height of the base is

. The height of the trapezoidalprism is 45 cm. The total volume of the tray is the sum of thevolumes of the two prisms.

ANSWER:

14,735 cm³

25. CHANGING DIMENSIONS A cereal companywants to increase the volume of each rectangularprism container by 25% without changing the base.Find the height of the new container if the originalhad a base of 8 inches by 2 inches and a height of12 inches. What would the height be if the surfacearea of the container increased by 25%?

SOLUTION:

Find the volume of the original container.

The volume of the new container is 125% of theoriginal container, with the same base dimensions.Use 1.25V and B to find h.

Next, find the surface area of the original container.

The surface area of the new container is 125% of theoriginal container, with the same base dimensions.Use 1.25S to find h.

inches

ANSWER:

15 in; 15.4 in.



Find the volume of each composite solid. Roundto the nearest tenth if necessary.

26.

SOLUTION:

The solid is a combination of two rectangular prisms.The base of one rectangular prism is 5 cm by 3 cmand the height is 11 cm. The base of the other prismis 4 cm by 3 cm and the height is 5 cm.

ANSWER:

225 cm3

27.

SOLUTION:

The solid is a combination of a rectangular prism anda right triangular prism. The total volume of the solidis the sum of the volumes of the two rectangularprisms.

ANSWER:

120 m3



28.

SOLUTION:

The solid is a combination of a rectangular prism anda half cylinder.

The volume of this combination shape is 713.1 yd³.

ANSWER:

713.1 in3

29. FOOD A cylindrical can of baked potato chips has aheight of 27 centimeters and a radius of 4centimeters. A new can is advertised as being 30%larger than the regular can. If both cans have thesame 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 smallercan, with the same radius.

The height of the new can will be 35.1 cm.

ANSWER:

35.1 cm



Find each measure to the nearest tenth.30. A cylindrical can has a volume of 363 cubic

centimeters. The diameter of the can is 9centimeters. What is the height?

SOLUTION:

ANSWER:

5.7 cm

31. A cylinder has a surface area of 144π square inchesand 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.

ANSWER:

678.6 in3



32. A rectangular prism has a surface area of 432 squareinches, 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 thebase of the prism.

Find the volume.

ANSWER:

576 in3

Find the volume of the solid formed by each net.

33.

SOLUTION:

The middle piece of the net is the front of the solid.The top and bottom pieces are the bases and thepieces on the ends are the side faces. This is atriangular prism. One leg of the base 14 cm and the hypotenuse 31.4cm. Use the Pythagorean Theorem to find the heightof the base.

The height of the prism is 20 cm. The volume V of a prism is V = Bh, where B is thearea of the base, h is the height of the prism.

ANSWER:

3934.9 cm3



34.

SOLUTION:

The circular bases at the top and bottom of the netindicate that this is a cylinder. If the middle piecewere a rectangle, then the prism would be right.However, since the middle piece is a parallelogram, itis oblique. The radius is 1.8 m, the height is 4.8 m, and the slantheight is 6 m. If two solids have the same height h and the samecross-sectional area B at every level, then they havethe same volume. So, the volume of a right prism andan oblique one of the same height and cross sectionalarea are same.

ANSWER:

48.9 m3

35. SOIL A soil scientist wants to determine the bulk densityof a potting soil to assess how well a specific plant willgrow in it. The density of the soil sample is the ratio of itsweight 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 isthe soil’s bulk density?b. Assuming that all other factors are favorable, how wellshould a plant grow in this soil if a bulk density of 0.018pound per square inch is desirable for root growth?Explain.c. If a bag of this soil holds 2.5 cubic feet, what is itsweight in pounds?

SOLUTION:

a. First calculate the volume of soil in the pot. Then dividethe weight of the soil by the volume.

The weight of the soil is the weight of the pot withsoil minus the weight of the pot.W = 20 – 5 = 15 lbs. The soil density is thus:

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant shouldgrow fairly well. c.

ANSWER:

a. 0.0019 lb / in3



b. The plant should grow well in this soil since the bulk

density of 0.0019 lb / in3 is close to the desired bulk

density of 0.0018 lb / in3.c. 8.3 lb

36. DESIGN Sketch and label (in inches) three differentdesigns for a dry ingredient measuring cup that holds1 cup. Be sure to include the dimensions in each

drawing. (1 cup ≈ 14.4375 in3)

SOLUTION: For any cylindrical container, we have the followingequation for volume:

The last equation gives us a relation between theradius and height of the cylinder that must be fulfilledto get the desired volume. First, choose a suitableradius, say 1.85 in, and solve for the height.

If we choose a height of say 4 in., then we can solvefor the radius.

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions andwe can solve for the third. Let = 2.25 in. and w =2.5 in.

ANSWER: Sample answers:



37. MODELING A cylindrical stainless steel column isused to hide a ventilation system in a new building.According to the specifications, the diameter of thecolumn can be between 30 centimeters and 95centimeters. The height is to be 500 centimeters.What is the difference in volume between the largestand smallest possible column? Round to the nearesttenth cubic centimeter.

SOLUTION:

The volume will be the highest when the diameter is95 cm and will be the lowest when it is 30 cm.That iswhen the radii are 47.5 cm and 15 cm respectively. Find the difference between the volumes.

ANSWER:

3,190,680.0 cm3

38. MULTISTEP Ryann is planning to build a sandcastle. She wants her castle to be 4 feet high, 4 feetwide, and 6 feet deep. She has asked her brotherJack to bring sand over to her building site. Theyeach have a bucket that is 8 inches in diameter and16 inches tall. Each trip takes Jack about 30seconds. a. After how long will Ryann have all of the sand shecould possibly need to complete her castle? b. Describe your solution process.c. What assumptions did you make?

SOLUTION:

a-b. The volume of the bucket is orabout 804 in³.If Jack carries two buckets at a time, then he cancarry 2(804) or 1608 in³. After converting thedimensions of the castle to inches, she finds themaximum volume of the castle is

. Therefore, it would takeJack or 104 trips, to round up for



the extra sand. If each trip takes him 30 seconds (or half a minute),

this would take him , or 52 minutes, toprovide her with the maximum amount of sand thatshe needs. c. The bucket is a cylinder. Ryann’s building site hadno sand to start with. Jack goes at the same pace forevery trip and doesn’t take any breaks. Each bucketis filled to the top and not overflowing with sand (orthe average bucket-full equals the full capacity of thebucket). Jack uses both buckets at the same time.Ryann never stops to help Jack. Ryann needs enoughsand to fill the entire volume.

ANSWER: a. 52 min.b. The volume of the bucket is 256π or about 804 in³.If Jack carries two buckets at a time, then he cancarry 1608 in³. The maximum volume of the castle is165,888 in³. Therefore, it would take Jack 104 trips,or 52 minutes, to provide her with the maximumamount of sand that she needs.c. The bucket is a cylinder. Ryann’s building site hadno sand to start with. Jack goes at the same pace forevery trip and doesn’t take any breaks. Each bucketis filled to the top and not overflowing with sand (orthe average bucket-full equals the full capacity of thebucket). Jack uses both buckets at the same time.Ryann never stops to help Jack. Ryann needs enoughsand to fill the entire volume.

39. Find the volume of the regular pentagonal prism bydividing it into five equal triangular prisms. Describethe base area and height of each triangular prism.

SOLUTION:

The base of the prism can be divided into 5 congruenttriangles of a base 8 cm and the corresponding height5.5 cm. So, the pentagonal prism is a combination of5 triangular prisms of height 10 cm. Find the basearea of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER:

1100 cm3; Each triangular prism has a base area of

or 22 cm2 and a height of 10 cm.



40. PATIOS Mr. Thomas is planning to remove an oldpatio and install a new rectangular concrete patio 20feet long, 12 feet wide, and 4 inches thick. Onecontractor bid $2225 for the project. A secondcontractor bid $500 per cubic yard for the new patioand $700 for removal of the old patio. Which is theless expensive option? Explain.

SOLUTION:

Convert all of the dimensions to yards.

20 feet = yd12 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 expensiveoption.

ANSWER:

Because 2.96 yd3 of concrete are needed, the secondcontractor is less expensive at $2181.50.

41. MULTIPLE REPRESENTATIONS In thisproblem, you will investigate right and obliquecylinders.a. GEOMETRIC Draw a right cylinder and anoblique cylinder with a height of 10 meters and adiameter of 6 meters.b. VERBAL A square prism has a height of 10meters and a base edge of 6 meters. Is its volumegreater than, less than, or equal to the volume ofthe cylinder? Explain.c. ANALYTICAL Describe which change affectsthe volume of the cylinder more: multiplying theheight by x or multiplying the radius by x. Explain.

SOLUTION:

a. The oblique cylinder should look like the rightcylinder (same height and size), except that it ispushed 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 lessthan x.

ANSWER:

a.



b. Greater than; a square with a side length of 6 mhas an area of 36 m2. A circle with a diameter of 6 m

has an area of 9π or 28.3 m2. Since the heights arethe same, the volume of the square prism is greater. c. Multiplying the radius by x ; since the volume isrepresented by π r 2 h, multiplying the height by xmakes the volume x times greater. Multiplying theradius by x makes the volume x 2 times greater,assuming x > 1.

42. ERROR ANALYSIS Franciso and Valerie eachcalculated the volume of an equilateral triangularprism with an apothem of 4 units and height of 5units. Is either of them correct? Explain yourreasoning.

SOLUTION:

Francisco is correct. Valerie incorrectly used asthe length of one side of the triangular base. Francisco used a different approach, but his solutionis correct. Francisco used the standard formula for the volumeof a solid, V = Bh. The area of the base, B, is one-half the apothem multiplied by the perimeter of thebase.

ANSWER:

Francisco; Valerie incorrectly used as thelength of one side of the triangular base. Franciscoused a different approach, but his solution is correct.

43. CHALLENGE The cylindrical can shown is used to

fill a container with liquid. It takes three full cans tofill the container. Describe possible dimensions of thecontainer if it is each of the following shapes. a. rectangular prismb. square prismc. triangular prism with a right triangle as the base

SOLUTION:

The volume of the can is 20π in3. It takes three fullcans to fill the container, so the volume of the

container is 60π in3. a. Choose some basic values for 2 of the sides, andthen determine the third side. Base: 3 by 5.

3 by 5 by 4π b. Choose some basic values for 2 of the sides, andthen determine the third side. Base: 5 by 5.

5 by 5 by c. Choose some basic values for 2 of the sides, andthen determine the third side. Base: Legs: 3 by 4.



3 by 4 by 10π

ANSWER:

Sample answers: a. 3 by 5 by 4π

b. 5 by 5 by c. base with legs measuring 3 in. and 4 in., height10π in.

44. WRITING IN MATH Write a helpful response tothe following question posted on an Internetgardening forum.I am new to gardening. The nursery will deliver atruckload of soil, which they say is 4 yards. Iknow that a yard is 3 feet, but what is a yard ofsoil? How do I know what to order?

SOLUTION:

The nursery means a cubic yard, which is 33 or 27cubic feet. Find the volume of your garden in cubicfeet and divide by 27 to determine the number ofcubic yards of soil needed.

ANSWER: Sample answer: The nursery means a cubic yard,

which is 33 or 27 cubic feet. Find the volume of yourgarden in cubic feet and divide by 27 to determine thenumber of cubic yards of soil needed.

45. OPEN-ENDED Draw and label a prism that has avolume of 50 cubic centimeters.

SOLUTION:

Choose 3 values that multiply to make 50. Thefactors of 50 are 2, 5, 5, so these are the simplestvalues to choose.

ANSWER:

Sample answer:



46. CONSTRUCT ARGUMENTS Determine whetherthe following statement is true or false. Explain.Two cylinders with the same height and the samelateral area must have the same volume.

SOLUTION:

The statement "Two cylinders with the same heightand the same lateral area must have the samevolume." is true. If two cylinders have the same height (h1 =

h2) and the same lateral area (L1 = L2), the circular

bases must have the same area.

The radii must also be equal.

ANSWER:

True; if two cylinders have the same height and thesame lateral area, the circular bases must have the

same area. Therefore, πr2h is the same for eachcylinder.

47. WRITING IN MATH How are the volumeformulas for prisms and cylinders similar? How arethey different?

SOLUTION:

Both formulas involve multiplying the area of the baseby the height. The base of a prism is a polygon, so theexpression representing the area varies, depending onthe type of polygon it is. The base of a cylinder is a

circle, so its area is πr2.

ANSWER:

Sample answer: Both formulas involve multiplying thearea of the base by the height. The base of a prism isa polygon, so the expression representing the areavaries, depending on the type of polygon it is. The

base of a cylinder is a circle, so its area is πr2.

48. The rectangular prism shown here has a square baseand a volume of 132.3 cubic inches.

What is the perimeter of the base? A 30 in.B 17.64 in.C 16.8 in.D 4.2 in.

SOLUTION:

Substitute the given information into the formula for arectangular prism and solve for x, one side of thesquare base.

If the length of one side of the square base is 4.2inches, then the perimeter is 4(4.2) or 16.8 inches.The correct choice is C.

ANSWER: C



49. An aquarium is a rectangular prism that is 20 incheslong, 1 foot wide, and 15 inches tall. Denise fills theaquarium using a container that holds 400 cubicinches of water. Assuming she always fills thecontainer completely, how many times will Deniseneed to pour water from the container into theaquarium?

SOLUTION: Begin by sketching a figure and label it with the giveninformation.

Find the volume of the prism:

She will need containers of 400 in³ waterto fill up the aquarium.

ANSWER: 9

50. Scott adds sand to the cylindrical container shownbelow so that the surface of the sand is 2 inchesbelow the top of the container.

Which of the following is the best estimate of thevolume of the sand in the container? A 127 in³B 269 in³C 318 in³D 445 in³E 1272 in³

SOLUTION: One way to find the volume of sand that fills thecylinder to a height 2 inches below the the top of thecontainer, subtract 2 inches from the height and findthe volume of this shorter cylinder.

The correct choice is C.

ANSWER: C



51. A cylindrical tank used for oil storage has a heightthat is half the length of its radius. If the volume ofthe tank is 1,122,360 ft³, what is the tank’s radius infeet? Round to the nearest tenth.

SOLUTION: Sketch a cylinder where the height is half the lengthof the radius.

Use these dimensions and the given volume to solvefor the length of the radius of the tank.

ANSWER: 89.4

52. The cylindrical can of juice shown here has a volumeof 300 cubic centimeters. What is the diameter of thecan in centimeters? Round to the nearest tenth.

A 6.2 cm B 3.1 cm C 9.8 cmD 30 cm

SOLUTION: Use the formula for the volume of a cylinder to findthe radius, then double the radius to find the height.

So, the diameter is about 6.18 centimeters. Thecorrect choice is A.

ANSWER: A

53. A red cube has an edge length of 2 inches. A bluecube has an edge length that is double that of the redcube. What is the volume of the blue cube?

SOLUTION: A blue cube has an edge length that is double that ofthe red cube or 2(2 inches) = 4 inches.

The volume of the blue cube is (4 inches)3 = 64 in.3

ANSWER:

64 in.3

54. MULTI-STEP Kara has a cylindrical pillar candlethat is 4 inches in diameter and 9 inches tall. Shemelts the candle and pours all of the wax into a



square mold that is 4 inches on each side.

a. To the nearest cubic inch, what is the volume ofthe pillar candle?

b. Write an equation that makes the volume of thepillar candle equal to the volume of a square candlewith 4-inch sides and an unknown height.

c. Solve the equation to find the height of the squarecandle.

d. If Kara wanted to make the square candle thesame height as the cylindrical candle (9 inches), howmuch more wax would she need?

SOLUTION: MULTI-STEP Kara has a cylindrical pillar candlethat is 4 inches in diameter and 9 inches tall. Shemelts the candle and pours all of the wax into asquare mold that is 4 inches on each side.

a. The volume of the pillar candle can be found usingthe formula for the volume of a cylinder.

V = πr2h

V = π(2 in.)2(9 in.)

V = 36π in.3

V ≈ 113 in.3

b. An equation with the volume of the pillar candleequal to the volume of a square candle with 4-inch

sides and an unknown height is 113 = (4)2h.

c. Solve the equation to find the height of the squarecandle.

113 = (4)2h

7.0625 = h The square candle is about 7 inches tall. d. To make the square candle the same height as the

cylindrical candle (9 inches), Kara would need 9(4)2

– 113 = 31 in.3of wax.

ANSWER:

a. 113 in.3

b. 113 = (4)2h.

c. 7 in.

d. 31 in.3



olqghu 5rxqgwrwkh€¦ · )lqgwkhyroxphrihdfkf\olqghu 5rxqgwrwkh qhduhvwwhqwk 62/87,21 $16:(5 iw...

Documents