Practice MCQ Questions and Answer on Volume and Surface Area

71.

A solid body is made up of a cylinder of radius r and height r, a cone of base radius r and height r fixed to the cylinder's one base and a hemisphere of radius r to its other base. The total volume of the body (given r = 2) is :

• (A) 8π
• (B) 16π
• (C) 32π
• (D) 64π

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 Total volume of the body : = Volume of the cylinder + Volume of the cone + Volume of the hemisphere $$\eqalign{ & = \pi {r^2}.\,r + \frac{1}{3}\pi {r^2}.\,r + \frac{2}{3}\pi {r^3} \cr & = 2\pi {r^3} \cr & = 2\pi {\left( 2 \right)^3} \cr & = 16\pi \cr} $$

72.

If the diameter of a sphere is 6 m, its hemisphere will have a volume of :

• (A) $$18\pi$$ m3
• (B) $$36\pi$$ m3
• (C) $$72\pi$$ m3
• (D) None of these

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 Volume of hemisphere : $$\eqalign{ & = \left( {\frac{2}{3}\pi \times 3 \times 3 \times 3} \right){m^3} \cr & = \left( {18\pi } \right){m^3} \cr} $$

73.

If the heights of two cones are in the ratio 7 : 3 and their diameters are in the ratio 6 : 7, what is the ratio of their volumes ?

• (A) 3 : 7
• (B) 4 : 7
• (C) 5 : 7
• (D) 12 : 7

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 Let the heights of two cones be 7x and 3x and their radii be 6y and 7y respectively Then, Ratio of volume : $$\eqalign{ & = \frac{{\frac{1}{3}\pi \times {{\left( {6y} \right)}^2} \times 7x}}{{\frac{1}{3}\pi \times {{\left( {7y} \right)}^2} \times 3x}} \cr & = \frac{{36 \times 7}}{{49 \times 3}} \cr & = \frac{{12}}{7}\,Or\,12:7 \cr} $$

74.

A metallic sheet is of rectangular shape with dimensions 48 m x 36 m. From each of its corners, a square is cut off so as to make an open box. If the length of the square is 8 m, the volume of the box (in m³) is:

• (A) 4830
• (B) 5120
• (C) 6420
• (D) 8960

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 Clearly, l = (48 - 16)m = 32 m, b = (36 -16)m = 20 m, h = 8 m. ∴ Volume of the box = (32 x 20 x 8) m3 = 5120 m3

75.

The diameter of the iron ball used for the shot-put game is 14 cm. It is melted and then a solid cylinder of height $$2\frac{1}{3}$$ cm is made. What will be the diameter of the base of the cylinder ?

• (A) 14 cm
• (B) $$\frac{14}{3}$$ cm
• (C) 28 cm
• (D) $$\frac{28}{3}$$ cm

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 Let the radius of the cylinder be R Then, $$\eqalign{ & \pi \times {R^2} \times \frac{7}{3} = \frac{4}{3}\pi \times 7 \times 7 \times 7 \cr & \Rightarrow {R^2} = \left( {\frac{{4 \times 7 \times 7 \times 7}}{3} \times \frac{3}{7}} \right) \cr & \Rightarrow {R^2} = 196 \cr & \Rightarrow {R^2} = {\left( {14} \right)^2} \cr & \Rightarrow {R^2} = 14{\text{ cm}} \cr} $$ ∴ Diameter = 2R = 28

76.

The surface area of a cube is 150 cm². Its volume is :

• (A) 64 $$\text{c}{\text{m}}^3$$
• (B) 125 $$\text{c}{\text{m}}^3$$
• (C) 150 $$\text{c}{\text{m}}^3$$
• (D) 216 $$\text{c}{\text{m}}^3$$

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 $$\eqalign{ & 6{a^2} = 150 \cr & \Rightarrow {a^2} = 25 \cr & \Rightarrow a = 5 \cr} $$ ∴ Volume : $${a^3} = {5^3} = 125\,{\text{ c}}{{\text{m}}^3}$$

77.

A closed box made of wood of uniform thickness has length, breadth and height 12 cm, 10 cm and 8 cm respectively. If the thickness of the wood is 1 cm, the inner surface area is :

• (A) 264 cm2
• (B) 376 cm2
• (C) 456 cm2
• (D) 696 cm2

Show Answer

 Internal length = (12 - 2) cm = 10 cm Internal breadth = (10 - 2) cm = 8 cm Internal height = (8 - 2) cm = 6 cm Inner surface area : = 2 [10 × 8 + 8 × 6 + 10 × 6] cm2 = (2 × 188) cm2 = 376 cm2

78.

If the volume of a sphere is divided by its surface area, the result is 27 cm. The radius of the sphere is :

• (A) 9 cm
• (B) 36 cm
• (C) 54 cm
• (D) 81 cm

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 $$\eqalign{ & \frac{{\frac{4}{3}\pi {r^3}}}{{4\pi {r^2}}} = 27 \cr & \Rightarrow r = 81\,cm \cr} $$

79.

The length, breadth and height of a cuboid are in the ratio 1 : 2 : 3. The length, breadth and height of the cuboid are increased by 100%, 200% and 200% respectively. Then the increase in the volume of the cuboid is :

• (A) 5 Times
• (B) 6 Times
• (C) 12 Times
• (D) 17 Times

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 Let the original length, breadth and height of the cuboid be x, 2x and 3x units respectively Then, original volume = (x × 2x × 3x) cu.units = 6x3 cu.units New length = 200% of x = 2x New breadth = 300% of 2x = 6x New height = 300% of 3x = 9x ∴ New volume : = (2x × 6x × 9x) cu.units = 108x3 cu.units Increase in volume : = (108x3 - 6x3) cu.units = (102x3) cu.units ∴ Required ratio : $$\eqalign{ & = \frac{{102{{\text{x}}^3}}}{{6{{\text{x}}^3}}} \cr & = 17{\text{ }}\left( {{\text{Times}}} \right) \cr} $$

80.

Water is poured into an empty cylindrical tank at a constant rate for 5 minutes. After the water has been poured into the tank. the depth of the water is 7 feet. The radius of the tank is 100 feet. Which of the following is the best approximation for the rate at which the water was poured into the tank ?

• (A) 140 cubic feet/sec
• (B) 440 cubic feet/sec
• (C) 700 cubic feet/sec
• (D) 2200 cubic feet/sec

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 Volume of water flown into the tank in 5 min : $$\eqalign{ & = \left( {\frac{{22}}{7} \times 100 \times 100 \times 7} \right){\text{cu}}{\text{.feet}} \cr & = 220000\,{\text{cu}}{\text{.feet}} \cr} $$ ∴ Rate of flow of water : $$\eqalign{ & = \left( {\frac{{220000}}{{5 \times 60}}} \right){\text{cu}}{\text{.feet/sec}} \cr & = 733.3 \approx 700\,{\text{cu}}{\text{.feet/sec}} \cr} $$