Practice MCQ Questions and Answer on Volume and Surface Area

81.

If the radius of the base and height of a cylinder and cone are each equal to r, and the radius of a hemisphere is also equal to r, then the volumes of the cone, cylinder and hemisphere are in the ratio ?

• (A) 1 : 2 : 3
• (B) 1 : 3 : 2
• (C) 2 : 1 : 3
• (D) 3 : 2 : 1

Show Answer

 Required ratio : = Volume of cone : Volume of cylinder : Volume of hemisphere $$\eqalign{ & = \frac{1}{3}\pi {r^2}:\pi {r^2}r:\frac{2}{3}\pi {r^3} \cr & = \frac{1}{3}:1:\frac{2}{3} \cr & = 1:3:2 \cr} $$

82.

The sum of perimeters of the six faces of a cuboid is 72 cm and the total surface area of the cuboid is 16 cm². Find the longest possible length that can be kept inside the cuboid :

• (A) 5.2 cm
• (B) 7.8 cm
• (C) 8.05 cm
• (D) 8.36 cm

Show Answer

 Sum of perimeters of the six faces : $$\eqalign{ & = 2\left[ {2\left( {l + b} \right) + 2\left( {b + h} \right) + 2\left( {l + h} \right)} \right] \cr & = 4\left( {2l + 2b + 2h} \right) \cr & = 8\left( {l + b + h} \right) \cr} $$ Total surface area $$ = 2\left( {lb + bh + lh} \right)$$ $$\eqalign{ & \therefore 8\left( {l + b + h} \right) = 72 \cr & \Rightarrow l + b + h = 9 \cr & 2\left( {lb + bh + lh} \right) = 16 \cr & \Rightarrow lb + bh + lh = 8 \cr} $$ Now, $${\left( {l + b + h} \right)^2} = {l^2} + {b^2} + {h^2} + 2$$     $$\left( {lb + bh + lh} \right)$$ $$\eqalign{ & \Rightarrow {\left( 9 \right)^2} = {l^2} + {b^2} + {h^2} + 16 \cr & \Rightarrow {l^2} + {b^2} + {h^2} = 81 - 16 \cr & \Rightarrow {l^2} + {b^2} + {h^2} = 65 \cr} $$ Required length : $$\eqalign{ & = \sqrt {{l^2} + {b^2} + {h^2}} \cr & = \sqrt {65} \cr & = 8.05\,cm \cr} $$

83.

The volumes of two cubes are in the ratio 8 : 27. The ratio of their surface areas is :

• (A) 2 : 3
• (B) 4 : 9
• (C) 12 : 9
• (D) None of these

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 Let their edges be a and b Then, $$\eqalign{ & \frac{{{a^3}}}{{{b^3}}} = \frac{8}{{27}} \cr & \Rightarrow {\left( {\frac{a}{b}} \right)^3} = {\left( {\frac{2}{3}} \right)^3} \cr & \Rightarrow \frac{a}{b} = \frac{2}{3} \cr & \Rightarrow \frac{{{a^2}}}{{{b^2}}} = \frac{4}{9} \cr & \Rightarrow \frac{{6{a^2}}}{{6{b^2}}} = \frac{4}{9}\,Or\,4:9 \cr} $$

84.

Water flows at the rate of 10 metres per minutes from a cylindrical pipe 5 mm in diameter. How long will it take to fill up a conical vessel whose diameter at the base is 40 cm and depth 24 cm ?

• (A) 48 minutes 15 seconds
• (B) 51 minutes 12 seconds
• (C) 52 minutes 1 seconds
• (D) 55 minutes

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 Volume flown in conical vessel : $$\eqalign{ & = \frac{1}{2}\pi \times {\left( {20} \right)^2} \times 24 \cr & = 3200\pi \cr} $$ Volume flown in 1 minute : $$\eqalign{ & = \left( {\pi \times \frac{{2.5}}{{10}} \times \frac{{2.5}}{{10}} \times 1000} \right) \cr & = 62.5\pi \cr} $$ ∴ Time taken : $$\eqalign{ & = \left( {\frac{{3200\pi }}{{62.5\pi }}} \right) \cr & = 51\operatorname{minutes} 12\operatorname{seconds} \cr} $$

85.

How many bricks, each measuring 25 cm × 11.25 cm × 6 cm, will be needed to build a wall 8 m × 6 m × 22.5 cm ?

• (A) 5600
• (B) 6000
• (C) 6400
• (D) 7200

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 $$\eqalign{ & {\text{Number of bricks :}} \cr & = \frac{{{\text{Volume of the wall}}}}{{{\text{Volume of 1 brick}}}} \cr & = \left( {\frac{{800 \times 600 \times 22.5}}{{25 \times 11.25 \times 6}}} \right) \cr & = 6400 \cr} $$

86.

The length of the longest rod that can be placed in a room of dimensions 10 m × 10 m × 5 m is :

• (A) 15$$\sqrt{3}$$ m
• (B) 15 m
• (C) 10$$\sqrt{2}$$ m
• (D) 5$$\sqrt{3}$$ m

Show Answer

 Required length : $$\eqalign{ & = \sqrt {{{\left( {10} \right)}^2} + {{\left( {10} \right)}^2} + {{\left( 5 \right)}^2}} \,m \cr & = \sqrt {225} \,m \cr & = 15\,m \cr} $$

87.

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

88.

How many small cubes, each of 96 cm surface area, can be formed from the material obtained by melting a larger cube of 384 cm surface area ?

• (A) 5
• (B) 8
• (C) 800
• (D) 8000

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 Let the length of each edge of small cube be a1 and that of larger cube be a2 Then, $$\eqalign{ & 6a_1^2 = 96\, \cr & \Rightarrow a_1^2 = 16 \cr & \Rightarrow {a_1} = 4\,and \cr & 6a_2^2 = 384 \cr & \Rightarrow a_2^2 = 64 \cr & \Rightarrow {a_2} = 8 \cr} $$ ∴ Number of cubes formed : $$\eqalign{ & = \frac{{{\text{Volume of larger cube}}}}{{{\text{Volume of smaller cube}}}} \cr & = \left( {\frac{{8 \times 8 \times 8}}{{4 \times 4 \times 4}}} \right) \cr & = 8 \cr} $$

89.

What is the volume in cubic cm of a pyramid whose area of the base is 25 sq cm and height 9 cm ?

• (A) 60 cm3
• (B) 75 cm3
• (C) 90 cm3
• (D) 105 cm3

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 Volume of pyramid :$$\eqalign{ & = \frac{1}{3} \times {\text{Area of base}} \times {\text{Height}} \cr & = \left( {\frac{1}{3} \times 25 \times 9} \right){\text{c}}{{\text{m}}^3} \cr & = 75\,{\text{c}}{{\text{m}}^3} \cr} $$

90.

A hall is 15 m long and 12 m broad. If the sum of the areas of the floor and the ceiling is equal to the sum of the areas of four walls, the volume of the hall is:

• (A) 720 m3
• (B) 900 m3
• (C) 1200 m3
• (D) 1800 m3

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 $$\eqalign{ & 2\left( {15 + 12} \right) \times h = 2\left( {15 \times 12} \right) \cr & \Rightarrow h = \frac{{180}}{{27}}m = \frac{{20}}{3}m \cr & \therefore {\text{Volume}} = \left( {15 \times 12 \times \frac{{20}}{3}} \right){m^3} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 1200\,{m^3} \cr} $$