Volume And Surface Area - Aptitude MCQ Questions and Solutions with Explanations

51

Water flows out through a circular pipe whose internal diameter is 2 cm, at the rate of 6 metres per second into a cylindrical tank, the radius of whose base is 60 cm. By how much will the level of water rise in 30 minutes ?

A) 2 m B) 3 m C) 4 m D) 5 m

Discuss

Answer & Solution

Answer: Option B

Solution:

Volume of water flown through the pipe in 30 min :
$$\eqalign{ & = \left[ {\left( {\pi \times 0.01 \times 0.01 \times 6} \right) \times 30 \times 60} \right]{m^3} \cr & = \left( {1.08\pi } \right){m^3} \cr} $$
Let the rise in level of water be h metres
Then,
$$\eqalign{ & \pi \times 0.6 \times 0.6 \times h = 1.08\pi \cr & \Rightarrow h = \left( {\frac{{1.08}}{{0.6 \times 0.6}}} \right) \cr & \Rightarrow h = 3\,m \cr} $$

52

What is the weight of water contained in a conical vessel 21 cm deep and 16 cm in diameter ?

A) 1.256 kg B) 1.408 kg C) 2.480 kg D) 3.875 kg

Discuss

Answer & Solution

Answer: Option B

Solution:

Volume of water :
$$\eqalign{ & = \left( {\frac{1}{3} \times \frac{{22}}{7} \times 8 \times 8 \times 21} \right){\text{ c}}{{\text{m}}^3} \cr & = 1408{\text{ c}}{{\text{m}}^3} \cr & = \left( {\frac{{1408}}{{1000}}} \right)kg \cr & = 1.408\,kg \cr} $$

53

If the ratio of volumes of two cones is 2 : 3 and the ratio of the radii of their bases is 1 : 2, then the ratio of their height will be :

A) 3 : 4 B) 4 : 3 C) 3 : 8 D) 8 : 3

Discuss

Answer & Solution

Answer: Option D

Solution:

let their radii be x and 2x, and their heights be h and H respectively
Then,
$$\eqalign{ & \frac{{\frac{1}{3} \times \pi \times {x^2} \times h}}{{\frac{1}{3} \times \pi \times {{\left( {2x} \right)}^2} \times H}} = \frac{2}{3} \cr & \Leftrightarrow \frac{h}{H} = \frac{2}{3} \times 4 \cr & \Leftrightarrow \frac{h}{H} = \frac{8}{3}\,Or\,8:3 \cr} $$

54

The volume of a sphere is $$2145\frac{{11}}{{21}}{\text{c}}{{\text{m}}^3}.$$ Its radius is equal to :

A) 7 cm B) 8 cm C) 9 cm D) None of these

Discuss

Answer & Solution

Answer: Option B

Solution:

$$\eqalign{ & \frac{4}{3} \times \frac{{22}}{7} \times {r^3} = \frac{{45056}}{{21}} \cr & \Rightarrow {r^3} = \left( {\frac{{45056}}{{21}} \times \frac{3}{4} \times \frac{7}{{22}}} \right) \cr & \Rightarrow {r^3} = 512 \cr & \Rightarrow r = \root 3 \of {512} \cr & \Rightarrow r = 8\,cm \cr} $$

55

A spherical ball of lead, 3 cm in diameter is melted and recast into three spherical ball. The diameter of two of these are 1.5 cm and 2 cm respectively. The diameter of the third ball is :

A) 2.5 cm B) 2.66 cm C) 3 cm D) 3.5 cm

Discuss

Answer & Solution

Answer: Option A

Solution:

Let the radius of the third ball be R cm
Then,
$$\frac{4}{3}\pi \times {\left( {\frac{3}{4}} \right)^3} + \frac{4}{3}\pi \times {\left( 1 \right)^3}$$ $$ + \frac{4}{3}\pi \times {R^3}$$ $$ = \frac{4}{3}\pi \times {\left( {\frac{3}{2}} \right)^3}$$
$$\eqalign{ & \Rightarrow \frac{{27}}{{64}} + 1 + {R^3} = \frac{{27}}{8} \cr & \Rightarrow {R^3} = \frac{{125}}{{64}} = \frac{{{{\left( 5 \right)}^3}}}{{{{\left( 4 \right)}^3}}} \cr & \Rightarrow R = \frac{5}{4} \cr} $$
∴ Diameter of the third ball :
$$ = 2R = \frac{5}{2}cm = 2.5\,cm$$

56

The ratio of the surface area of a sphere and the curved surface area of the cylinder circumscribing the sphere is :

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

Discuss

Answer & Solution

Answer: Option A

Solution:

Let the radius of the sphere be r
Then, radius of the cylinder = r
Height of the cylinder = 2r
Surface area of sphere = $$4\pi {{\text{r}}^2}$$
Surface area of the cylinder = $$2\pi {\text{r}}(2r) = 4\pi {{\text{r}}^2}$$
∴ Required ratio :
= $$4\pi {{\text{r}}^2}$$ : $$4\pi {{\text{r}}^2}$$
= 1 : 1

57

A hemispherical bowl of internal radius 12 cm contains liquid. This liquid is to be filled into cylindrical container of diameter 4 cm and height 3 cm. The number of containers that is necessary to empty the bowl is :

A) 80 B) 96 C) 100 D) 112

Discuss

Answer & Solution

Answer: Option B

Solution:

Volume of hemispherical bowl :
$$ = \left( {\frac{2}{3} \times \pi \times 12 \times 12 \times 12} \right)c{m^3}$$
Volume of 1 cylindrical container :
$$ = \left( {\pi \times 2 \times 2 \times 3} \right)c{m^3}$$
∴ Number of containers required :
$$\eqalign{ & = \frac{2}{3} \times \frac{{12 \times 12 \times 12}}{{2 \times 2 \times 3}} \cr & = 96 \cr} $$

58

Length of each edge of a regular tetrahedron is 1 cm. It volume is :

A) $$\frac{{\sqrt 3 }}{{12}}{\text{ }}c{m^3}$$ B) $$\frac{1}{4}\sqrt 3 {\text{ }}c{m^3}$$ C) $$\frac{{\sqrt 2 }}{6}{\text{ }}c{m^3}$$ D) $$\frac{1}{{12}}\sqrt 2 {\text{ }}c{m^3}$$

Discuss

Answer & Solution

Answer: Option D

Solution:

Length of each edge of a regular tetrahedron = 1 cm
Volume of regular tetrahedron :
$$\eqalign{ & = \frac{{{a^3}}}{{6\sqrt 2 }}{\text{ c}}{{\text{m}}^3} \cr & = \frac{1}{{6\sqrt 2 }} \cr & = \frac{{\sqrt 2 }}{{6\sqrt 2 \times \sqrt 2 }}{\text{ c}}{{\text{m}}^3} \cr & = \frac{{\sqrt 2 }}{{12}}{\text{ Or }}\frac{1}{{12}}\sqrt 2 {\text{ c}}{{\text{m}}^3} \cr} $$

59

The base of a right prism is a trapezium whose lengths of two parallels sides are 10 cm and 6 cm and distance between them is 5 cm. If the heights of the prism is 8 cm, its volume is :

A) 320 cm³ B) 300 cm³ C) 310 cm³ D) 300.5 cm³

Discuss

Answer & Solution

Answer: Option A

Solution:

Length of parallel sides of prism = 10 cm and 6 cm
Height of prism = 8 cm
∴ Volume of prism :
$$\eqalign{ & = \frac{1}{2}\left( {10 + 6} \right) \times 5 \times 8 \cr & = \frac{1}{2} \times 16 \times 5 \times 8 \cr & = 320{\text{ c}}{{\text{m}}^3} \cr} $$

60

A rectangular water reservoir contains 42000 litres of water. If the length of reservoir is 6 m and breadth of the reservoir is 3.5 m, then the depth of the reservoir will be :

A) 2 m B) 5 m C) 6 m D) 8 m

Discuss

Answer & Solution

Answer: Option A

Solution:

Volume of the reservoir = 42000 litres = 42 m³
Let the depth of the reservoir be h metres
then,
$$\eqalign{ & 6 \times 3.5 \times h = 42 \cr & Or,\,h = \frac{{42}}{{6 \times 3.5}} = 2\,m \cr} $$