Volume and Surface Area problems
- 1. The diameters of two cones are equal. If their slant heights be in the ratio of 5 : 7, then find the ratio of their curved surface areas.?
Options- A. 25 : 7
- B. 25 : 49
- C. 5 : 49
- D. 5 : 7 Discuss
Correct Answer: 5 : 7
Explanation:
Given, l1/l2 = 5/7
Now, curved surface area of the first cone = ? rl1
and curved surface area of second cone = ?rl2
? Ratio = ?rl1/?rl2 = l1/l2 = 5 : 7
- 2. The radius of the base of a right circular cone is increased by 15% keeping the height fixed. The volume of the cone will be increased by.?
Options- A. 30%
- B. 31%
- C. 32.25%
- D. 34.75% Discuss
Correct Answer: 32.25%
Explanation:
Let the fixed height of a right circular cone is h and initial radius is r
Then, initial volume of cone, V1 = (1/3)?r2h
After increasing 15% radius of a cone = (r + 3r/20) = 23r/20
New volume become, V2 = (1/3)?(23/20)2r2h
? Increasing percentage = [(V2 - V1) / V1] x 100
= {[(1/3)?r2h] / [(1/3)?r2h]} {(23/20)2 - 1} x 100
= (23/20 + 1)(23/20 - 1) x 100
= 43/20 x 3/20 x 100 = 32.25%
- 3. What is the diameter of the largest circle lying on the surface of a sphere of surface area 616 sq cm?
Options- A. 14 cm
- B. 10.5 cm
- C. 7 cm
- D. 3.5 cm Discuss
Correct Answer: 14 cm
Explanation:
Surface area of sphere = 6616 cm2
4?r2 = 6166
?r2 = (6166 x 7)/(4 x 22)
? r2 = 7 x 7
? r = 7 cm
? Diameter of largest circle lying on sphere = 2 x r = 2 x 7 = 14 cm
- 4. The diameter of the Moon is approximately one-fourth of the diameter of the Earth. What is the ratio (approximate) of their volumes?
Options- A. 4 : 16
- B. 1 : 64
- C. 1 : 4
- D. 1 : 128 Discuss
Correct Answer: 1 : 64
Explanation:
Given that, the diameter of moon is approximately one-fourth of the diameter of Earth.
Let radius on moon = r
Then, radius of Earth = 4r
? Volume of Moon/Volume of Earth
= [(4/3)?r3] / [(4/3)?(4r)3]
= [r3] / [64r3] = 1/64 = 1 : 64
- 5. A sphere and a hemisphere have the same surface area. The ratio of their volumes is?
Options- A. ?3/4 : 1
- B. 3?3/4 : 1
- C. ?3/8 : 1
- D. 3?3/8 : 1 Discuss
Correct Answer: 3?3/8 : 1
Explanation:
According to the question,
Surface area of sphere = Surface area of hemisphere
4?r12 =3?r22
? r1/r2 = ?3/2
? Ratio in volume = [(4/3)?r13] / [(4/3)?r23]
= 3?3/8 : 1
- 6. Find the number of lead balls of diameter 2 cm each, that can be made from a sphere of diameter 16 cm.?
Options- A. 2048
- B. 2055
- C. 2058
- D. None of the above Discuss
Correct Answer: None of the above
Explanation:
Radius of the sphere = 16/2 = 8 cm
Volume of the sphere = (4/3) x ? x 8 x 8 x 8 cm3
Radius of each lead ball = 2/2 = 1 cm
Volume of each lead ball = Volume of sphere / Volume of lead ball
= (4/3) ? x 1 x 1 x 1 = 4?/3 cm3
? Number of lead balls = [(4/3) x ? x 8 x 8 x 8 x 3] / [4 ?]
= 8 x 8 x 8 = 512
- 7. A hemispherical bowl has 3.5 cm radius. It is to be painted inside as well as outside. Find the cost of painting it at the rate of ? 5 per 10 sq cm.?
Options- A. ? 50
- B. ? 81
- C. ? 56
- D. ? 77 Discuss
Correct Answer: ? 77
Explanation:
Curved surface area of the hemisphere = 2?r2
= 2 x (22/7) x (7/2) x (7/2) = 77 sq
As bowl is to painted inside and outside.
? Total surface to be painted = 77 x 2 = 154 sq cm
? Cost of painting 154 sq cm = (5/10) x 154 = 1/2 x 154 = ? 77
- 8. If the surface area of a sphere is 616 sq cm, what is its volume?
Options- A. 4312/3 cm3
- B. 4102/3 cm3
- C. 1257 cm3
- D. 1023 cm3 Discuss
Correct Answer: 4312/3 cm3
Explanation:
Curved surface area of the sphere = 4?r2
or 616 = 4?r2
? ?r2 = 616/4 = 154
? r2 = (154 x 7) / 22 = 49
? r = ?49 = 7 cm
? Volume of the sphere = (4/3)?r3
= (4/3) x (22/7) x 7 x 7 x 7
= 4312 / 3 cm3
- 9. If the ratio of the diameters of two spheres is 3: 5, then what is the ratio of their surface areas?
Options- A. 9 : 25
- B. 9 : 10
- C. 3 : 5
- D. 27 : 125 Discuss
Correct Answer: 9 : 25
Explanation:
Let the diameter's of two sphere are d1 and d2, respectively.
? Ratio of their surface areas = 4?r12/4?r22
= (2r1)2/(2r2)2 = d12/d22
= (d1/d2)2 = (3/5)2 = 9/25 = 9 : 25
- 10. If 64 identical small spheres are made out of a big sphere of diameter 8 cm, what is surface area of each small sphere?
Options- A. ? cm2
- B. 2 ? cm2
- C. 4 ? cm2
- D. 8 ? cm2 Discuss
Correct Answer: 4 ? cm2
Explanation:
Volume of small spheres
= Volume of bigger sphere / Number of small spheres = [(4/3)?(4)3] / 64
= [(4/3) x ? x 4 x 4 x 4] / 64
= 4/3 ? cm3
Let radius of small sphere be r
? 4/3?r3 = 4?/3
? r2 = 1 cm
Now, surface area of small sphere = 4?r2 = 4? cm2
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