)LQGWKHVXUIDFHDUHDRIHDFKVSKHUHRU U 62/87,21

Find the surface area of each sphere orhemisphere. Round to the nearest tenth.

1.

SOLUTION:

ANSWER:

1017.9 m2

2.

SOLUTION:

ANSWER:

461.8 in2

3. sphere: area of great circle = 36π yd2

SOLUTION:

We know that the area of a great circle is . Findr.

Now find the surface area.

ANSWER:

452.4 yd2

4. hemisphere: circumference of great circle ≈ 26 cm

SOLUTION:

We know that the circumference of a great circle is .

The area of a hemisphere is one-half the area of thesphere plus the area of the great circle.

ANSWER:

161.4 cm2

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Find the volume of each sphere or hemisphere.Round to the nearest tenth.

5. sphere: radius = 10 ft

SOLUTION:

ANSWER:

4188.8 ft3

6. hemisphere: diameter = 16 cm

SOLUTION:

ANSWER:

1072.3 cm3

7. hemisphere: circumference of great circle = 24π m

SOLUTION:

We know that the circumference of a great circle is. Find r.

Now find the volume.

ANSWER:

3619.1 m3

8. sphere: area of great circle = 55π in2

SOLUTION:

We know that the area of a great circle is . Findr.

Now find the volume.

ANSWER:

1708.6 in3


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9. BASKETBALL Basketballs used in professional

games must have a circumference of 29 inches.

What is the surface area of a basketball used in aprofessional game?

SOLUTION:

We know that the circumference of a great circle is2πr . Find r.

Find the surface area.

ANSWER:

277.0 in2

Find the surface area of each sphere orhemisphere. Round to the nearest tenth.

10.

SOLUTION:

ANSWER:

50.3 ft2

11.

SOLUTION:

ANSWER:

113.1 cm2

12.

SOLUTION:

ANSWER:

109.0 mm2


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13.

SOLUTION:

ANSWER:

680.9 in2

14. sphere: circumference of great circle = 2π cm

SOLUTION:

We know that the circumference of a great circle is2πr. Find r.

ANSWER:

12.6 cm2

15. sphere: area of great circle ≈ 32 ft2

SOLUTION:

We know that the area of a great circle is πr2. Find r.

ANSWER:

128 ft2

16. hemisphere: area of great circle ≈ 40 in2

SOLUTION:

We know that the area of a great circle is πr2. Find r.

Substitute for r2 in the surface area formula.

ANSWER:

120 in2


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17. hemisphere: circumference of great circle = 15π mm

SOLUTION:

ANSWER:

530.1 mm2

PRECISION Find the volume of each sphere orhemisphere. Round to the nearest tenth.

18.

SOLUTION:

The volume V of a hemisphere is or

, where r is the radius.

The radius is 5 cm.

ANSWER:

261.8 ft3

19.

SOLUTION:

The volume V of a sphere is , where r is

the radius. The radius is 1 cm.

ANSWER:

4.2 cm3

20. sphere: radius = 1.4 yd

SOLUTION:


the radius.

ANSWER:

11.5 yd3


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21. hemisphere: diameter = 21.8 cm

SOLUTION:

The radius is 10.9 cm. The volume V of a hemisphere

is or , where r is the

radius.

ANSWER:

2712.3 cm3

22. sphere: area of great circle = 49π m2

SOLUTION:

The area of a great circle is .


the radius.

ANSWER:

1436.8 m3

23. sphere: circumference of great circle ≈ 22 in.

SOLUTION:

The circumference of a great circle is .


the radius.

ANSWER:

179.8 in3

24. hemisphere: circumference of great circle ≈ 18 ft

SOLUTION:

The circumference of a great circle is .



ANSWER:

49.2 ft3


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25. hemisphere: area of great circle ≈ 35 m2

SOLUTION:

The area of a great circle is .



ANSWER:

77.9 m3

26. FISH A puffer fish is able to “puff up” whenthreatened by gulping water and inflating its body.The puffer fish at the right is approximately a spherewith a diameter of 5 inches. Its surface area wheninflated is about 1.5 times its normal surface area.What is the surface area of the fish when it is notpuffed up? Refer to the photo on Page 823.

SOLUTION:

Find the surface area of the non-puffed up fish, orwhen it is normal.

ANSWER:

about 52.4 in2


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27. ARCHITECTURE The Reunion Tower in Dallas,Texas, is topped by a spherical dome that has asurface area of approximately 13,924π square feet.What is the volume of the dome? Round to thenearest tenth.Refer to the photo on Page 823.

SOLUTION:

Find r.

Find the volume.

ANSWER:

860,289.5 ft3

28. TREE HOUSE The spherical tree house, or treesphere, has a diameter of 10.5 feet. Its volume is 1.8times the volume of the first tree sphere that wasbuilt. What was the diameter of the first tree sphere?Round to the nearest foot.

SOLUTION:

The volume of the spherical tree house is 1.8 timesthe volume of the first tree sphere.

The diameter is 4.3(2) or about 9 ft.

ANSWER:

9 ft


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PERSEVERANCE Find the surface area andthe volume of each solid. Round to the nearesttenth.

29.

SOLUTION:

To find the surface area of the figure, calculate thesurface area of the cylinder (without the bases),hemisphere (without the base), and the base, and addthem.

To find the volume of the figure, calculate the volumeof the cylinder and the hemisphere and add them.

ANSWER:

276.5 in2; 385.4 in.3

30.

SOLUTION:

To find the surface area of the figure, we will need tocalculate the surface area of the prism, the area ofthe top base, the area of the great circle, and one-halfthe surface area of the sphere. First, find the surface area of the prism and subtractthe area of the top base.

This covers everything but the top of the figure. Thecorners of the top base can be found by subtractingthe area of the great circle from the area of the base.

Next, find the surface area of the hemisphere.

Now, find the total surface area.


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The volume of the figure is the volume of the prismminus the volume of the hemisphere.

ANSWER:

798.5 cm2; 1038.2 cm3

31. TOYS The spinning top is a composite of a cone anda hemisphere.

a. Find the surface area and the volume of the top.Round to the nearest tenth.b. If the manufacturer of the top makes anothermodel with dimensions that are one-half of thedimensions of this top, what are its surface area andvolume?

SOLUTION:

a. Use the Pythagorean Theorem to find the slantheight.

Lateral area of the cone:

Surface area of the hemisphere:

b. Use the Pythagorean Theorem to find the slantheight.

Lateral area of the cone:

Surface area of the hemisphere:


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ANSWER:

a. 594.6 cm2 ; 1282.8 cm3

b. 148.7 cm2 ; 160.4 cm3

32. BALLOONS A spherical helium-filled balloon with adiameter of 30 centimeters can lift a 14-gram object.Find the size of a balloon that could lift a person whoweighs 65 kilograms. Round to the nearest tenth.

SOLUTION:

1 kg = 1000 g.Form a proportion.Let x be the unknown.

The balloon would have to have a diameter ofapproximately 139,286 cm.

ANSWER:

The balloon would have to have a diameter ofapproximately 139,286 cm.

Use sphere S to name each of the following.

33. a chord

SOLUTION:

A chord of a sphere is a segment that connects anytwo points on the sphere.

ANSWER:

34. a radius

SOLUTION:

A radius of a sphere is a segment from the center toa point on the sphere.

Sample answer:

ANSWER:

Sample answer:

35. a diameter

SOLUTION:

A diameter of a sphere is a chord that contains thecenter.

ANSWER:


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36. a tangent

SOLUTION:

A tangent to a sphere is a line that intersects thesphere in exactly one point. Line l intersects thesphere at A.

ANSWER:

line l

37. a great circle

SOLUTION:

If the circle contains the center of the sphere, theintersection is called a great circle.

SANSWER:

S38. DIMENSIONAL ANALYSIS Which has greater

volume: a sphere with a radius of 2.3 yards or acylinder with a radius of 4 feet and height of 8 feet?

SOLUTION:

2.3 yards = 2.3(3) or 6.9 feet

Since 1376 ft3 > 402 ft3, the sphere has the greatervolume.

ANSWER:

the sphere

39. INFORMAL PROOF A sphere with radius r can

be thought of as being made up of a large number ofdiscs or thin cylinders. Consider the disc shown thatis x units above or below the center of the spheres.Also consider a cylinder with radius r and height 2rthat is hollowed out by two cones of height and radiusr.

a. Find the radius of the disc from the sphere in termsof its distance x above the sphere's center. (Hint:Use the Pythagorean Theorem.)b. If the disc from the sphere has a thickness of yunits, find its volume in terms of x and y.c. Show that this volume is the same as that of thehollowed-out disc with thickness of y units that is xunits above the center of the cylinder and cone. d. Since the expressions for the discs at the sameheight are the same, what guarantees that thehollowed-out cylinder and sphere have the samevolume?e . Use the formulas for the volumes of a cylinder anda cone to derive the formula for the volume of thehollowed-out cylinder and thus, the sphere.

SOLUTION: a. Let d be the radius of the disc at a height x. Fromthe diagram below, we can see how to use thePythagorean Theorem to express d in terms of x andr.


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b. If each disc has a thickness of y, then the discsform tiny cylinders, and we can use the volumeformula and the expression found in part a.

c. For a cylinder the radius of each disc is always r,at any height x. The volume of discs in the cylinder isthus:

. For the cone, the height of the top half is r and theradius is r. By similar triangles, at any height x abovethe center, the radius of the disc at that height is alsox.

The volume of a disc at height x is thus:

Rearranging the terms and taking the difference wehave:

d. We have shown that each of the figures have thesame height and that they have the same crosssectional area at each height, which implies, byCavlieri's Principle that the two objects have thesame volume. e . The volume of the cylinder is:

The volume of the cones is:

From parts c and d, the volume of the sphere is justthe difference of these two;

ANSWER:

a.

b.

c. The volume of the disc from the cylinder is πr2y or

πyr2. The volume of the disc from the two cones is

πx2y or πyx2. Subtract the volumes of the discs from

the cylinder and cone to get πyr2 – πyx2, which is theexpression for the volume of the disc from the sphereat height x.d. Cavalieri’s Principle

e . The volume of the cylinder is πr2(2r) or 2πr3. The

volume of one cone is , so thevolume of the double napped cone is

. Therefore, the volume of thehollowed out cylinder, and thus the sphere, is

.

PERSEVERANCE Describe the number andtypes of planes that produce reflectionalsymmetry in each solid. Then describe theangles of rotation that produce rotationalsymmetry in each solid.

40. a sphere


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SOLUTION:

There is an infinite number of planes that producereflectional symmetry for spheres. Any plane thatpasses through the center will produce reflectionalsymmetry. Any plane that does not pass through thecenter will not produce symmetry.These are all symmetric

These are not:

The angle of rotation is the angle through which apreimage is rotated to form the image. A sphere canbe rotated at any angle, and the resulting image willbe identical to the preimage.

ANSWER:

There are infinitely many planes that producereflectional symmetry as long as they pass throughthe origin. Any angle of rotation will producerotational symmetry.

41. a hemisphere

SOLUTION:

Any plane of symmetry must pass through the origin.Take a look at a few examples.

symmetric


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Symmetric

Not symmetric

Not symmetric We see that the only planes that produce reflectionalsymmetry are vertical planes that pass through theorigin. The angle of rotation is the angle through which apreimage is rotated to form the image. A hemispherecan be rotated at any angle around the vertical axis,and the resulting image will be identical to thepreimage. A hemisphere rotated about any other axiswill not be symmetric.

Symmetric

Not Symmetric

ANSWER:

There are infinitely many planes that producereflectional symmetry as long as they are verticalplanes that pass through the center. Only rotationabout an axis through which the center isperpendicular to the base will produce rotationalsymmetry through infinitely many angles.


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CHANGING DIMENSIONS A sphere has aradius of 12 centimeters. Describe how eachchange affects the surface area and the volumeof the sphere.

42. The radius is multiplied by 4.

SOLUTION:

9216 ÷ 576 = 16

The surface area is multiplied by 42 or 16. The

volume is multiplied by 43 or 64.

ANSWER:

The surface area is multiplied by 42 or 16. The

volume is multiplied by 43 or 64.

43. The radius is divided by 3.

SOLUTION:

576 ÷ 64 = 9

The surface area is divided by 32 or 9. The volume is

divided by 33 or 27.

ANSWER:

The surface area is divided by 32 or 9. The volume is

divided by 33 or 27.

44. DESIGN A standard juice box holds 8 fluid ounces.a. Sketch designs for three different juice containersthat will each hold 8 fluid ounces. Label dimensions incentimeters. At least one container should be

cylindrical. (Hint: 1 fl oz ≈ 29.57353 cm3) b. For each container in part a, calculate the surface

area to volume (cm2 per fl oz) ratio. Use these ratiosto decide which of your containers can be made forthe lowest materials cost. What shape containerwould minimize this ratio, and would this container be


11-4 Spheres

the cheapest to produce? Explain your reasoning.

SOLUTION: a. Sample answer: Choose two cylindrical containersone with height 9 cm and one with height 7 cm, and athird container that is a cube.The radii of the cylindrical containers can be foundusing the formula for volume. For 9 cm height:

For 7 cm height:

We can similarly solve for the length of the cube:

b. First calculate the surface area of each of thesolids in part a. For cylinders:

For the cube:

The ratios of surface area to volume are thus:Container A

Container B


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Container C

Container B has the lowest material cost. If weconsider the ratio of surface area to volume for asphere we get:

A sphere has the lowest surface area to volume ratioof any three dimensional object, but the cost withtrying to manufacture spherical containers would begreatly increased, since special machines would haveto be used.

ANSWER: a. Sample answers:

b. Sample answers: Container A, ≈ 27.02 cm2 per fl

oz; Container B, ≈26.48 cm2 per fl oz; Container C, ≈

28.69 cm2 per fl oz; Of thesethree, Container B can be made for the lowestmaterials cost. The lower the surface area to volumeratio, the less packaging used for each fluid ounce ofjuice it holds. A spherical container with r = 3.837 cmwould minimize this cost since it would have the leastsurface area to volume ratio of any shape, ≈ 23.13cm per fl oz. However, a spherical container wouldlikely be more costly to manufacture than arectangular container since specially made machinerywould be necessary.

45. MODELING A cube has a volume of 216 cubicinches. Find the volume of a sphere that iscircumscribed about the cube. Round to the nearesttenth.

SOLUTION:

Since the volume of the cube is 216 cubic inches, thelength of each side is 6 inches and so, the length of a

diagonal on a face is inches.


11-4 Spheres

Use the length of the one side (BC) and this diagonal(AB) to find the length of the diagonal joining twoopposite vertices of the cube (AC). The triangle thatis formed by these sides is a right triangle. Apply thePythagorean Theorem.

The sphere is circumscribed about the cube, so thevertices of the cube all lie on the sphere. Therefore,line AC is also the diameter of the sphere. The radius

is inches. Find the volume.

ANSWER:

587.7 in3

46. ERROR ANALYSIS Diego says that the lateralarea of a basketball is 29.5 square inches. Rodneysays that you cannot calculate the lateral area of asphere. Who is correct? Explain your reasoning.

SOLUTION:

Since a sphere has no base, then it can't have anylateral faces. Therefore, you can only calculate itssurface area. Rodney is correct.

ANSWER:

Rodney is correct. Since a sphere has no lateralfaces, you can only calculate its surface area.


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47. CHALLENGE Sketch a sphere showing twoexamples of great circles. Sketch another sphereshowing two examples of circles formed by planesintersecting the sphere that are not great circles.

SOLUTION:

The great circles need to contain the center of thesphere. The planes that are not great circles need tonot contain the center of the sphere.

ANSWER:

Sample answer:

48. WRITING IN MATH Write a ratio comparing thevolume of a sphere with radius r to the volume of acylinder with radius r and height 2r. Then describewhat the ratio means.

SOLUTION:

The ratio is 2:3.The volume of the sphere is two thirds the volume ofthe cylinder.

ANSWER:

; The volume of the sphere is two thirds the

volume of the cylinder.


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49. An artist is planning an exhibit in a hemisphericalbuilding with a diameter of 40 feet. The artist wantsto cover the inside of the hemisphere (floor andceiling) with black paint. Which of the following is thebest estimate of the surface area that will bepainted?

A 2513 ft²B 3770 ft²C 5027 ft²D 15,080 ft²E 16,755 ft²

SOLUTION: Find the surface area of a hemisphere with a radiusof 20 feet and combine it with the area of anothercircle (the floor) with the same radius.

The correct choice is B.

ANSWER: B

50. What is the circumference of a sphere that has avolume of 288π cubic meters?

A 144π mB 36π mC 12π mD 6π m

SOLUTION: To find the circumference of the sphere, you firstneed to determine its radius. Use the given volume tosolve for the radius of the sphere.

Now, we know the radius of the sphere is 6 so thecircumference would be Thecorrect choice is C.

ANSWER: C


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51. Adam has a spherical candle with a radius of 2inches. He melts the candle completely and pours thesoft wax into a mold in the shape of a rectangularprism. The dimensions of the mold are shown here.

Which of the following best describes how the meltedwax will fill the mold? A The wax will fill slightly more than one-third of themold.B The wax will fill the mold more than halfway.C The wax will fill the mold almost perfectly.D The wax will fill the mold, and there will be a lot ofleftover wax.

SOLUTION: First, find the volume of the spherical candle.

Then, find the volume of the rectangular prism mold.

, so the candle will fill a littlemore than half the mold. The correct choice is B.

ANSWER: B

52. A spherical disco ball has a surface area of 169πsquare centimeters.

a. Find the radius of the ball.b. Find the volume of the ball.

SOLUTION:

a. Surface area = 4πr2

169π cm2 = 4πr2

169 = 4r2

42.25 = r2

6.5 = r The radius is 6.5 cm. b.

The volume is cm3.

ANSWER: a. 6.5 cm

b. cm3

53. MULTI-STEP Alice bought two sphericalornaments with radii of 2 centimeters and 4centimeters. Find the following.

a. the volume of the first ornament b. the volume of the second ornament c. the surface area of the first ornament d. the surface area of the second ornament e . the ratio of the volume of the first ornament to thevolume of the second ornament

f. the ratio of the surface area of the first ornament tothe surface area of the second ornamentthe surface area of the second ornament

SOLUTION:


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a. the volume of the first ornament =

cm3

b. the volume of the second ornament =

cm3

c. the surface area of the first ornament =

cm2

d. the surface area of the second ornament =

cm2

e . the ratio of the volume of the first ornament to thevolume of the second ornament is 32 : 256 or 1 : 8. f. the ratio of the surface area of the first ornament tothe surface area of the second ornament is 16 : 64 or1 : 4.

ANSWER:

a. cm3

b. cm3

c. cm2

d. cm2

e . 1 : 8. f. 1 : 4.

54. A ball has a diameter of 18 centimeters. A cylinderholds the ball exactly. What is the volume of thecylinder?

SOLUTION: The cylinder that holds the ball exactly has a height of18 cm and a diameter of 18 cm.

The volume of the cylinder is 1458π cm3.

ANSWER:

1458π cm3

55. MULTI-STEP The length of the equator on a globeis 44 inches. Answer the following questions,rounding to the nearest tenth.

a. What is the diameter of the globe?b. What is the radius of the globe?c. What is the surface area of the globe?d. What is the volume of the globe?

SOLUTION: The length of the equator is a circumference of aglobe, which in this problem is 44 inches.

a. The diameter of the globe is the circumferencedivided by pi or about 14.0 in.b. The radius of the globe is half the diameter orabout 7.0 in. c. The surface area of the globe is 4 times pi times

the radius squared, or about 616.2 in.2 d. The volume of the globe is four-thirds pi times the

radius cubed, or about 1438.5 in.3

ANSWER: a. 14.0 in.b. 7.0 in.

c. 616.2 in.2

d. 1438.5 in.3


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