## Volume and Surface Area Questions and Answers Part-5

1.A cistern of capacity 8000 litres measures externally 3.3 m by 2.6 m by 1.1 m and its walls are 5 cm thick. The thickness of the bottom is :
a) 90 cm
b) 1 dm
c) 1 m
d) 1.1 m

Explanation: Let the thickness of the bottom be x cm
$$\left[ {\left( {330 - 10} \right) \times \left( {260 - 10} \right) \times \left( {110 - x} \right)} \right]$$       $$= 8000 \times 1000$$
$$\Rightarrow 320 \times 250 \times \left( {110 - x} \right) =$$       $$8000 \times 1000$$
\eqalign{ & \left( {110 - x} \right) = \frac{{8000 \times 1000}}{{320 \times 250}} \cr & \left( {110 - x} \right) = 100 \cr & x = 10\,cm \cr & x = 1\,dm \cr}

2. 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

Explanation:
\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}

3. A solid cube just gets completely immersed in water when a 0.2 kg mass is placed on it. If the mass is removed, the cube is 2 cm above the water level. What is the length of each side of the cube ?
a) 6 cm
b) 8 cm
c) 10 cm
d) 12 cm

Explanation: Let the length of each side of the cube be a cm
Then, Volume of the part of cube outside water = Volume of the mass placed on it
2a2 = 0.2 × 1000
2a2 = 200
a2 = 100
a = 10

4.The volume of a right circular cylinder, 14 cm in height is equal to that of a cube whose edge is 11 cm. The radius of the base of the cylinder is :
a) 5.2 cm
b) 5.5 cm
c) 11 cm
d) 22 cm

Explanation: Volume of the cylinder :
= Volume of the cube
= (11)3 cm3
= 1331 cm3
Let the radius of the base be r cm
\eqalign{ & \frac{{22}}{7} \times {r^2} \times 14 = 1331 \cr & {r^2} = \frac{{1331}}{{44}} = \frac{{121}}{4} \cr & r = \frac{{11}}{2} \cr & r = 5.5 \cr}

5. If two cylinders of equal volumes have their heights in the ratio 2 : 3, then the ratio if their radii is :
a) $$\sqrt 6 :\sqrt 3$$
b) $$\sqrt 5 :\sqrt 3$$
c) $$2 : 3$$
d) $$\sqrt 3 :\sqrt 2$$

Explanation: Let their heights be 2h and 3h and radii be r and R respectively
\eqalign{ & \pi {r^2}\left( {2h} \right) = \pi {R^2}\left( {3h} \right) \cr & \frac{{{r^2}}}{{{R^2}}} = \frac{3}{2} \cr & \Rightarrow \frac{r}{R} = \frac{{\sqrt 3 }}{{\sqrt 2 }}\,i.e.,\sqrt 3 :\sqrt 2 \cr}

6. The slant height of a right circular cone is 10 m and its height is 8 m. Find the area of its curved surface.
a) 30π m2
b) 40π m2
c) 60π m2
d) 80π m2

Explanation:
\eqalign{ & l = 10\,m,h = 8\,m \cr & So, \cr & r = \sqrt {{l^2} + {h^2}} \cr & \,\,\,\, = \sqrt {{{\left( {10} \right)}^2} - {{\left( 8 \right)}^2}} \cr & \,\,\,\, = 6\,m \cr}
Curved surface area :
\eqalign{ & = \pi rl \cr & = \left( {\pi \times 6 \times 10} \right){m^2} \cr & = 60\pi \,{m^2} \cr}

7. A solid metallic right circular cylinder of base diameter 16 m and height 2 cm is melted and recast into a right circular cone of height three times that of the cylinder. Find the curved surface area of the cone. [Use $$\pi$$ = 3.14]
a) 196.8 cm2
b) 228.4 cm2
c) 251.2 cm2
d) None of these

Explanation: Let the radius of the cone be r cm
\eqalign{ & \pi \times {\left( 8 \right)^2} \times 2 = \frac{1}{3} \times \pi \times {r^2} \times 6 \cr & \Rightarrow r = 8 \cr}
Slant height,
\eqalign{ & l = \sqrt {{r^2} + {h^2}} \cr & \,\,\, = \sqrt {{8^2} + {6^2}} \cr & \,\,\, = \sqrt {100} \cr & \,\,\, = 10\,cm \cr}
Curved surface area of cone :
\eqalign{ & = \pi rl \cr & = \left( {3.14 \times 8 \times 10} \right){\text{ c}}{{\text{m}}^2} \cr & = 251.2{\text{ c}}{{\text{m}}^2} \cr}

8. 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

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

9. A sphere and a cube have equal surface area. The ratio of the volume of the sphere to that of the cube is :
a) $$\sqrt \pi :\sqrt 6$$
b) $$\sqrt 2 :\sqrt \pi$$
c) $$\sqrt \pi :\sqrt 3$$
d) $$\sqrt 6 :\sqrt \pi$$

Explanation:
\eqalign{ & 4\pi {R^2} = 6{a^2} \cr & \frac{{{R^2}}}{{{a^2}}} = \frac{3}{{2\pi }} \cr & \frac{R}{a} = \frac{{\sqrt 3 }}{{\sqrt {2\pi } }} \cr}
\eqalign{ & \frac{{{\text{Volume of spere}}}}{{{\text{Volume of cube}}}} \cr & = \frac{{\frac{4}{3}\pi {R^3}}}{{{a^3}}} \cr & = \frac{4}{3}\pi {\left( {\frac{R}{a}} \right)^3} \cr & = \frac{4}{3}\pi \frac{{3\sqrt 3 }}{{2\pi \sqrt {2\pi } }} \cr & = \frac{{2\sqrt 3 }}{{\sqrt {2\pi } }} \cr & = \frac{{\sqrt {12} }}{{\sqrt {2\pi } }} \cr & = \frac{{\sqrt 6 }}{{\sqrt \pi }} \cr & \text{or, }\sqrt 6 :\sqrt \pi \cr}

10. Some solid metallic right circular cones, each with radius of the base 3 cm and height 4 cm, are melted to form a solid sphere of radius 6 cm. The number of right circular cones is :
a) 6
b) 12
c) 24
d) 48

\eqalign{ & = \left( {\frac{4}{3}\pi \times {6^3}} \right){\text{c}}{{\text{m}}^{\text{3}}} \cr & = \left( {288\pi } \right){\text{c}}{{\text{m}}^{\text{3}}} \cr}
\eqalign{ & = \left( {\frac{1}{3}\pi \times {3^2} \times 4} \right){\text{c}}{{\text{m}}^{\text{3}}} \cr & = \left( {12\pi } \right){\text{c}}{{\text{m}}^{\text{3}}} \cr}
\eqalign{ & = \frac{{288\pi }}{{12\pi }} \cr & = 24 \cr}