Find the Surface Area of Each Sphere Or Hemisphere. Round to the Nearest Tenth

Home , Surface, Surface (mathematics), Surface (topology), Surface area

Find the surface area of each sphere or 3. sphere: area of great circle = 36π yd2 hemisphere. Round to the nearest tenth. SOLUTION: We know that the area of a great circle is . Find r.

SOLUTION:

Now find the surface area.

ANSWER: 1017.9 m2 ANSWER: 452.4 yd2

4. hemisphere: circumference of great circle ≈ 26 cm

SOLUTION: 2. We know that the circumference of a great circle is . SOLUTION:

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

ANSWER: 461.8 in2

ANSWER: 161.4 cm2

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Find the volume of each sphere or hemisphere. 7. hemisphere: circumference of great circle = 24π m Round to the nearest tenth. SOLUTION: 5. sphere: radius = 10 ft We know that the circumference of a great circle is SOLUTION: . Find r.

Now find the volume.

ANSWER: 4188.8 ft3

6. hemisphere: diameter = 16 cm ANSWER: SOLUTION: 3619.1 m3

8. sphere: area of great circle = 55π in 2

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

ANSWER: 1072.3 cm3

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 a professional game? 11.

SOLUTION: SOLUTION: We know that the circumference of a great circle is 2πr . Find r.

ANSWER: 113.1 cm2 Find the surface area.

12.

SOLUTION: ANSWER: 277.0 in2

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

ANSWER: 10. 109.0 mm2 SOLUTION:

ANSWER: 50.3 ft2

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15. sphere: area of great circle ≈ 32 ft2

SOLUTION: 2 13. We know that the area of a great circle is πr . Find r.

SOLUTION:

ANSWER: 680.9 in2 ANSWER: 14. sphere: circumference of great circle = 2π cm 128 ft2

SOLUTION: 16. hemisphere: area of great circle ≈ 40 in2 We know that the circumference of a great circle is 2πr. Find r. SOLUTION: We know that the area of a great circle is πr2. Find r.

Substitute for r2 in the surface area formula.

ANSWER: 12.6 cm2

ANSWER: 120 in2

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

SOLUTION:

19.

SOLUTION:

The volume V of a sphere is , where r is the radius.

The radius is 1 cm.

ANSWER: 530.1 mm2

PRECISION Find the volume of each sphere or hemisphere. Round to the nearest tenth. ANSWER: 4.2 cm3

20. sphere: radius = 1.4 yd

18. SOLUTION:

SOLUTION: The volume V of a sphere is , where r is The volume V of a hemisphere is or the radius.

, where r is the radius.

The radius is 5 cm. ANSWER: 11.5 yd3

ANSWER: 261.8 ft3

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21. hemisphere: diameter = 21.8 cm 23. sphere: circumference of great circle ≈ 22 in.

SOLUTION: SOLUTION: The radius is 10.9 cm. The volume V of a hemisphere The circumference of a great circle is .

is or , where r is the

radius.

The volume V of a sphere is , where r is the radius.

ANSWER: 2712.3 cm3

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

SOLUTION: ANSWER: 3 The area of a great circle is . 179.8 in 24. hemisphere: circumference of great circle ≈ 18 ft

SOLUTION: The circumference of a great circle is . The volume V of a sphere is , where r is the radius.

The volume V of a hemisphere is or

ANSWER: , where r is the radius. 1436.8 m3

ANSWER: 49.2 ft3

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25. hemisphere: area of great circle ≈ 35 m2 26. FISH A puffer fish is able to “puff up” when threatened by gulping water and inflating its body. SOLUTION: The puffer fish at the right is approximately a sphere The area of a great circle is . with a diameter of 5 inches. Its surface area when inflated is about 1.5 times its normal surface area. What is the surface area of the fish when it is not puffed up? Refer to the photo on Page 823.

SOLUTION: The volume V of a hemisphere is or

, where r is the radius.

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

ANSWER: 77.9 m3

ANSWER: about 52.4 in2

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27. ARCHITECTURE The Reunion Tower in Dallas, 28. TREE HOUSE The spherical tree house, or tree Texas, is topped by a spherical dome that has a sphere, has a diameter of 10.5 feet. Its volume is 1.8 surface area of approximately 13,924π square feet. times the volume of the first tree sphere that was What is the volume of the dome? Round to the built. What was the diameter of the first tree sphere? nearest tenth. Round to the nearest foot. Refer to the photo on Page 823.

SOLUTION: Find r.

SOLUTION:

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

ANSWER: 860,289.5 ft3

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

ANSWER: 9 ft

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PERSEVERANCE Find the surface area and the volume of each solid. Round to the nearest tenth.

30. 29. SOLUTION: SOLUTION: To find the surface area of the figure, calculate the To find the surface area of the figure, we will need to surface area of the cylinder (without the bases), calculate the surface area of the prism, the area of hemisphere (without the base), and the base, and add the top base, the area of the great circle, and one-half them. the surface area of the sphere.

First, find the surface area of the prism and subtract the area of the top base.

To find the volume of the figure, calculate the volume This covers everything but the top of the figure. The of the cylinder and the hemisphere and add them. corners of the top base can be found by subtracting the area of the great circle from the area of the base.

Next, find the surface area of the hemisphere.

ANSWER: Now, find the total surface area. 276.5 in2; 385.4 in.3

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Lateral area of the cone: The volume of the figure is the volume of the prism minus the volume of the hemisphere.

Surface area of the hemisphere:

ANSWER: 798.5 cm2; 1038.2 cm3

31. TOYS The spinning top is a composite of a cone and b. Use the Pythagorean Theorem to find the slant a hemisphere. height.

Lateral area of the cone:

a. Find the surface area and the volume of the top. Round to the nearest tenth. b. If the manufacturer of the top makes another model with dimensions that are one-half of the dimensions of this top, what are its surface area and Surface area of the hemisphere: volume?

SOLUTION: a. Use the Pythagorean Theorem to find the slant height. eSolutions Manual - Powered by Cognero Page 10 11-4 Spheres

Use sphere S to name each of the following.

ANSWER: 33. a chord a. 594.6 cm2 ; 1282.8 cm3 SOLUTION: b. 148.7 cm2 ; 160.4 cm3 A chord of a sphere is a segment that connects any 32. BALLOONS A spherical helium-filled balloon with a two points on the sphere. diameter of 30 centimeters can lift a 14-gram object. Find the size of a balloon that could lift a person who weighs 65 kilograms. Round to the nearest tenth. ANSWER: SOLUTION: 1 kg = 1000 g. Form a proportion. 34. a radius Let x be the unknown. SOLUTION: A radius of a sphere is a segment from the center to a point on the sphere.

Sample answer:

The balloon would have to have a diameter of ANSWER: approximately 139,286 cm. Sample answer:

ANSWER: 35. a diameter The balloon would have to have a diameter of approximately 139,286 cm. SOLUTION: A diameter of a sphere is a chord that contains the center.

ANSWER:

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36. a tangent be thought of as being made up of a large number of discs or thin cylinders. Consider the disc shown that SOLUTION: is x units above or below the center of the spheres. A tangent to a sphere is a line that intersects the Also consider a cylinder with radius r and height 2r sphere in exactly one point. Line l intersects the that is hollowed out by two cones of height and radius sphere at A. r.

ANSWER: line l

37. a great circle

SOLUTION: a. Find the radius of the disc from the sphere in terms If the circle contains the center of the sphere, the of its distance x above the sphere's center. (Hint: Use the Pythagorean Theorem.) intersection is called a great circle. b. If the disc from the sphere has a thickness of y

units, find its volume in terms of x and y. S c. Show that this volume is the same as that of the hollowed-out disc with thickness of y units that is x ANSWER: units above the center of the cylinder and cone. d. Since the expressions for the discs at the same S height are the same, what guarantees that the hollowed-out cylinder and sphere have the same 38. DIMENSIONAL ANALYSIS Which has greater volume? volume: a sphere with a radius of 2.3 yards or a e. Use the formulas for the volumes of a cylinder and cylinder with a radius of 4 feet and height of 8 feet? a cone to derive the formula for the volume of the hollowed-out cylinder and thus, the sphere. SOLUTION: SOLUTION: 2.3 yards = 2.3(3) or 6.9 feet a. Let d be the radius of the disc at a height x. From the diagram below, we can see how to use the Pythagorean Theorem to express d in terms of x and r.

3 3 Since 1376 ft > 402 ft , the sphere has the greater volume.

ANSWER: the sphere

39. INFORMAL PROOF A sphere with radius r can eSolutions Manual - Powered by Cognero Page 12 11-4 Spheres

b. If each disc has a thickness of y, then the discs form tiny cylinders, and we can use the volume formula and the expression found in part a.

The volume of the cones is:

From parts c and d, the volume of the sphere is just the difference of these two; c. For a cylinder the radius of each disc is always r, at any height x. The volume of discs in the cylinder is thus: .

For the cone, the height of the top half is r and the radius is r. By similar triangles, at any height x above the center, the radius of the disc at that height is also ANSWER: x. 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 the expression for the volume of the disc from the sphere at height x. d. Cavalieri’s Principle The volume of a disc at height x is thus: e. The volume of the cylinder is πr2(2r) or 2πr3. The

volume of one cone is , so the Rearranging the terms and taking the difference we volume of the double napped cone is have: . Therefore, the volume of the hollowed out cylinder, and thus the sphere, is

d. We have shown that each of the figures have the . same height and that they have the same cross PERSEVERANCE Describe the number and sectional area at each height, which implies, by Cavlieri's Principle that the two objects have the types of planes that produce reflectional same volume. symmetry in each solid. Then describe the angles of rotation that produce rotational e. The volume of the cylinder is: symmetry in each solid. 40. a sphere eSolutions Manual - Powered by Cognero Page 13 11-4 Spheres

SOLUTION: There is an infinite number of planes that produce reflectional symmetry for spheres. Any plane that passes through the center will produce reflectional symmetry. Any plane that does not pass through the center will not produce symmetry. These are all symmetric

The angle of rotation is the angle through which a preimage is rotated to form the image. A sphere can be rotated at any angle, and the resulting image will be identical to the preimage.

ANSWER: There are infinitely many planes that produce reflectional symmetry as long as they pass through the origin. Any angle of rotation will produce rotational symmetry.

41. a hemisphere

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

These are not:

symmetric

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ANSWER: There are infinitely many planes that produce reflectional symmetry as long as they are vertical planes that pass through the center. Only rotation about an axis through which the center is perpendicular to the base will produce rotational symmetry through infinitely many angles. Symmetric

Not symmetric

We see that the only planes that produce reflectional symmetry are vertical planes that pass through the origin.

The angle of rotation is the angle through which a preimage is rotated to form the image. A hemisphere can be rotated at any angle around the vertical axis, and the resulting image will be identical to the preimage. A hemisphere rotated about any other axis will not be symmetric.

Symmetric

Not Symmetric

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CHANGING DIMENSIONS A sphere has a 43. The radius is divided by 3. radius of 12 centimeters. Describe how each SOLUTION: change affects the surface area and the volume of the sphere. 42. The radius is multiplied by 4.

SOLUTION:

576 ÷ 64 = 9

9216 ÷ 576 = 16

The surface area is divided by 32 or 9. The volume is divided by 33 or 27.

The surface area is multiplied by 42 or 16. The ANSWER: volume is multiplied by 43 or 64. The surface area is divided by 32 or 9. The volume is divided by 33 or 27. ANSWER: The surface area is multiplied by 42 or 16. The 44. DESIGN A standard juice box holds 8 fluid ounces. volume is multiplied by 43 or 64. a. Sketch designs for three different juice containers that will each hold 8 fluid ounces. Label dimensions in centimeters. 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 ratios to decide which of your containers can be made for the lowest materials cost. What shape container would minimize this ratio, and would this container be eSolutions Manual - Powered by Cognero Page 16 11-4 Spheres

the cheapest to produce? Explain your reasoning.

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

For 9 cm height:

b. First calculate the surface area of each of the For 7 cm height: solids in part a. For cylinders:

We can similarly solve for the length of the cube:

For the cube:

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

Container B eSolutions Manual - Powered by Cognero Page 17 11-4 Spheres

Container C

Container B has the lowest material cost. If we consider the ratio of surface area to volume for a sphere we get:

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 these three, Container B can be made for the lowest materials cost. The lower the surface area to volume ratio, the less packaging used for each fluid ounce of juice it holds. A spherical container with r = 3.837 cm would minimize this cost since it would have the least surface area to volume ratio of any shape, ≈ 23.13 cm per fl oz. However, a spherical container would

likely be more costly to manufacture than a rectangular container since specially made machinery would be necessary.

45. MODELING A cube has a volume of 216 cubic A sphere has the lowest surface area to volume ratio inches. Find the volume of a sphere that is of any three dimensional object, but the cost with circumscribed about the cube. Round to the nearest trying to manufacture spherical containers would be tenth. greatly increased, since special machines would have to be used. SOLUTION: Since the volume of the cube is 216 cubic inches, the ANSWER: length of each side is 6 inches and so, the length of a a. Sample answers: diagonal on a face is inches.

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

SOLUTION: Since a sphere has no base, then it can't have any lateral faces. Therefore, you can only calculate its surface area. Rodney is correct.

ANSWER: Rodney is correct. Since a sphere has no lateral faces, you can only calculate its surface area. Use the length of the one side (BC) and this diagonal (AB) to find the length of the diagonal joining two opposite vertices of the cube (AC). The triangle that is formed by these sides is a right triangle. Apply the Pythagorean Theorem.

The sphere is circumscribed about the cube, so the vertices 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

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47. CHALLENGE Sketch a sphere showing two 48. WRITING IN MATH Write a ratio comparing the examples of great circles. Sketch another sphere volume of a sphere with radius r to the volume of a showing two examples of circles formed by planes cylinder with radius r and height 2r. Then describe intersecting the sphere that are not great circles. what the ratio means.

SOLUTION: SOLUTION: The great circles need to contain the center of the sphere. The planes that are not great circles need to not contain the center of the sphere.

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

ANSWER:

; The volume of the sphere is two thirds the volume of the cylinder.

ANSWER: Sample answer:

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49. An artist is planning an exhibit in a hemispherical 50. What is the circumference of a sphere that has a building with a diameter of 40 feet. The artist wants volume of 288π cubic meters? to cover the inside of the hemisphere (floor and ceiling) with black paint. Which of the following is the A 144π m best estimate of the surface area that will be B 36π m painted? C 12π m D 6π m A 2513 ft² B 3770 ft² SOLUTION: C 5027 ft² To find the circumference of the sphere, you first D 15,080 ft² need to determine its radius. Use the given volume to E 16,755 ft² solve for the radius of the sphere.

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

Now, we know the radius of the sphere is 6 so the circumference would be The correct choice is C. The correct choice is B. ANSWER: ANSWER: C B

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51. Adam has a spherical candle with a radius of 2 52. A spherical disco ball has a surface area of 169π inches. He melts the candle completely and pours the square centimeters. soft wax into a mold in the shape of a rectangular prism. The dimensions of the mold are shown here. a. Find the radius of the ball. b. Find the volume of the ball.

SOLUTION: a. Surface area = 4πr2 169π cm2 = 4πr2 169 = 4r2 Which of the following best describes how the melted 42.25 = r2 wax will fill the mold? 6.5 = r

A The wax will fill slightly more than one-third of the The radius is 6.5 cm. mold. B The wax will fill the mold more than halfway. b. C The wax will fill the mold almost perfectly. D The wax will fill the mold, and there will be a lot of leftover wax.

SOLUTION: First, find the volume of the spherical candle. The volume is cm3.

ANSWER: a. 6.5 cm

b. cm3

Then, find the volume of the rectangular prism mold. 53. MULTI-STEP Alice bought two spherical ornaments with radii of 2 centimeters and 4 centimeters. Find the following.

a. the volume of the first ornament

b. the volume of the second ornament , so the candle will fill a little more than half the mold. The correct choice is B. c. the surface area of the first ornament

ANSWER: d. the surface area of the second ornament B e. the ratio of the volume of the first ornament to the volume of the second ornament

f. the ratio of the surface area of the first ornament to the surface area of the second ornament

SOLUTION: eSolutions Manual - Powered by Cognero Page 22 11-4 Spheres

a. the volume of the first ornament = 55. MULTI-STEP The length of the equator on a globe 3 is 44 inches. Answer the following questions, cm rounding to the nearest tenth.

b. the volume of the second ornament = a. What is the diameter of the globe? cm3 b. What is the radius of the globe? c. What is the surface area of the globe? c. the surface area of the first ornament = d. What is the volume of the globe? cm2 SOLUTION: The length of the equator is a circumference of a d. the surface area of the second ornament = globe, which in this problem is 44 inches. cm2 a. The diameter of the globe is the circumference e. the ratio of the volume of the first ornament to the divided by pi or about 14.0 in. volume of the second ornament is 32 : 256 or 1 : 8. b. The radius of the globe is half the diameter or about 7.0 in. f. the ratio of the surface area of the first ornament to c. The surface area of the globe is 4 times pi times the surface area of the second ornament is 16 : 64 or the radius squared, or about 616.2 in.2 1 : 4. d. The volume of the globe is four-thirds pi times the 3 ANSWER: radius cubed, or about 1438.5 in.

a. cm3 ANSWER: a. 14.0 in. 3 b. cm b. 7.0 in. 2 c. cm c. 616.2 in.2 2 d. cm d. 1438.5 in.3 e. 1 : 8. f. 1 : 4.

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

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

The volume of the cylinder is 1458π cm3.

ANSWER: 1458π cm3 eSolutions Manual - Powered by Cognero Page 23