# Problems on Volumes and Surface Area

Q21. Find the area of the iron sheet required to prepare a cone 20 cm height with base radius 15 cm.

(a) 1126.60 cm^{2}

(b) 1365.25 cm^{2}

(c) 1750.35 cm^{2}

(d) 1885.71 cm^{2}

**Answer:** (d) 1885.71 cm^{2}

Here r = 15 cm

h = 20 cm

∴ Slant height l = | = | = | = | = 25 cm | ||||

√r^{2} + h^{2} |
√15^{2} + 20^{2} |
√225 + 400 | √625 |

Area of the sheet = πrl + πr^{2}

= πr (l + r)

= ^{22}/_{7} X 15 (25 + 15)

= ^{22}/_{7} X 15 x 40

= 1885.71 cm^{2}

Q22. A rectangular water tank is open at the top. Its capacity is 24m^{3}. Its length and breadth are 4 m and 3 m respectively. Ignoring the thickness of the material used for building the tank, the total cost of painting the ineer and outer surfaces of the tank at Rs. 10 per m^{2} is

(a) Rs. 650

(b) Rs. 750

(c) Rs. 800

(d) Rs. 900

**Answer:** (c) Rs. 800

Let the height of the tank be x meter

∴ Volume of the tank = 4 X 3 X x m^{3}

⇒ 24 m^{3} = 12x m^{3}

⇒ x = 2 m

Area of the surface to be painted = 2 [2(l+b) X h + (lXb)]

= 2 [2 X (4+3) X 2 + 4 X 3 ]

= 2 [28 + 12] = 80 m^{2}

∴ Cost of painting = Rs. 80 X 10 = Rs. 800

Q23. What will be the area of an iron sheet which can be used to form a cone having base radius 7 cm and height 24 cm ?

(a) 704 cm^{2}

(b) 712 cm^{2}

(c) 726 cm^{2}

(d) 740 cm^{2}

**Answer:** (a) 704 cm^{2}

r = 7 cm h = 24 cm

∴ l = √(r^{2}+h^{2}) = √(7^{2}+24^{2}) = √(49+576) = √(625) = 25 cm

Required area = Total surface area of the cone

= πr(l+r) = ^{22}/_{7} X 7(25+7) = 22 X 32 = 704 cm^{2}

Q24. The ratio of the radius and height of a cone is 5:12. Its volume is 314 ^{2}/_{7} cc. Its slant height is.

(a) 13 cm

(b) 15 cm

(c) 16 cm

(d) 21 cm

**Answer:** (a) 13 cm

Let the radius and height of the cone be 5x, 12x

∴ Volume = ^{1}/_{3}πr^{2}h

⇒ 314 | 2 | = | 1 | X | 22 | X (5x)^{2} X 12x |

7 | 3 | 7 |

⇒ | 2200 | = | 22 | X 25x^{2} X 12x |

7 | 21 |

⇒ x^{3} = |
2200 X 21 |

7 X 22 X 25 X 12 |

⇒ x^{3} = 1

⇒ x = 1

∴ r = 5 h = 12

∴ l = √(r^{2}+h^{2}) = √(5^{2}+12^{2}) = √(25+144) = 13cm

Q25. If the ratio of volumes of two cones is 2:3 and the ratio of the raddi of their bases is 1:2, then the ratio of their heights will be

(a) 5:4

(b) 5:3

(c) 7:3

(d) 8:3

**Answer:** (d) 8:3

Here | V_{1} |
= | 2 |

V_{2} |
3 |

R_{1} |
= | 1 |

R_{2} |
2 |

Now | V_{1} |
= | ^{1}/_{3} π R_{1}^{2}H_{1} |

V_{2} |
^{1}/_{3} π R_{2}^{2}H_{2} |

Now | 2 | = | R_{1}^{2}H_{1} |

3 | R_{2}^{2}H_{2} |

⇒ | 2 | = ( | R_{1} |
)^{2} |
H_{1} |

3 | R_{2} |
H_{2} |

⇒ | 2 | = ( | R_{1} |
)^{2} |
H_{1} |

3 | R_{2} |
H_{2} |

⇒ | 2 | = ( ½ )^{2} |
H_{1} |

3 | H_{2} |

⇒ | H_{1} |
= | 2 | X 4 = | 8 |

H_{2} |
3 | 3 |

∴ H_{1} : H_{2} = 8 : 3

Q26. The volume of a hemisphere of radius 7 cm is

(a) 712.5 cm^{3}

(b) 718.6 cm^{3}

(c) 723.6 cm^{3}

(d) 728.3 cm^{3}

**Answer:** (b) 718.6 cm^{3}

Volume = ^{2}/_{3} πr^{3}

= | 2 | X | 22 | X 7^{3} |

3 | 7 |

= | 44 X 49 |

3 |

= 718.6 cm^{3}

Q27. How many balls of diameter 0.4 cm each are formed a cuboid of measuring (4cm X 5cm X 6cm) ?

(a) 3512

(b) 3580

(c) 3620

(d) 3675

**Answer:** (b) 3580

Volume of the cuboid = 4cm X 5cm X 6cm = 120 cm^{3}

Volume of each ball = ^{4}/_{3}πr^{3}

= | 4 | X | 22 | X | 0.4 | X | 0.4 | X | 0.4 |

3 | 7 | 2 | 2 | 2 |

= | 0.704 |

21 |

= 0.03352

Number of balls = | 120 |

0.03352 |

= 3580 (approx)

Q28. The volume of a sphere is 2145 ^{11}/_{21} cm^{3}. Its diameter is

(a) 16 cm

(b) 18 cm

(c) 21 cm

(d) 23 cm

**Answer:** (a) 16 cm

Volume = ^{4}/_{3}πr^{3}

⇒ | 45056 | = | 4 | X | 22 | X | r^{3} |

21 | 3 | 7 |

⇒ r^{3} = |
45056 | X | 3 X 7 |

21 | 4 X 22 |

⇒ r^{3} = 512

⇒ r^{3} = 8^{3}

⇒ r = 8

∴ Diameter = 2 X 8 = 16 cm

Q29. A cone of height 15 cm and base diameter 30 cm is carved out of a wooden sphere of radius 15 cm. The percentage of wasted wood is

(a) 61%

(b) 65%

(c) 72%

(d) 75%

**Answer:** (d) 75%

Volume = ^{4}/_{3}πr^{3}

= | 4 | X | 22 | X 15^{3} |

3 | 7 |

Volume of cone =^{1}/_{3}πr^{2}h = |
1 | X | 22 | X 15^{2} X 15 = |
1 | X | 22 | X 15^{3} |

3 | 7 | 3 | 7 |

Wasted wood = | 4 | X | 22 | X 15^{3} - |
1 | X | 22 | X 15^{3} |

3 | 7 | 3 | 7 |

= | 22 | X 15^{3} = π15^{3} |

7 |

Percentage of wasted wood = | π X 15 ^{3} |
X 100 % |

^{4}/_{3} X 15^{3} |

= ¾ X 100% = 75%

Q30. 12 sphere of the same size are made by melting a solid cylinder of 16 cm diameter and 2 cm height. The diameter of each sphere is

(a) 4 cm

(b) 6 cm

(c) 7 cm

(d) 8 cm

**Answer:** (a) 4 cm

Here H = 2 cm R = 8 cm

Volume of the cylinder = πR^{2}h = π8^{2} X 2 = 128π cm^{3}

Let, the radius of each sphere be r

then

12 X | 4 | πr^{3} = 128π |

3 |

⇒ r^{3} = |
128π X 3 |

48π |

⇒ r^{3} = 8

⇒ r = 2

∴ Diameter = 2 X 2 = 4 cm

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