Practice MCQ Questions and Answer on Volume and Surface Area

41.

The breadth of a room is twice its height and half its length. The volume of the room is 512 cu.m. The length of the room is :

• (A) 16 m
• (B) 18 m
• (C) 20 m
• (D) 32 m

Show Answer

 Let the height of the room be x metres Then, breadth = 2x metres and length = 4x metres ∴ Volume of the room : = (4x × 2x × x) m3 = (8x3) m3 8x3 = 512 ⇒ x3 = 64 ⇒ x = 4 ∴ Length of the room is : = 4x = (4 × 4) = 16 m

42.

A swimming pool 9 m wide and 12 m long is 1 m deep on the shallow side and 4 m deep on the deeper side. It volume is :

• (A) 208 m3
• (B) 270 m3
• (C) 360 m3
• (D) 408 m3

Show Answer

 Volume : $$\eqalign{ & = \left[ {12 \times 9 \times \left( {\frac{{1 + 4}}{2}} \right)} \right]{m^3} \cr & = \left( {12 \times 9 \times 2.5} \right){m^3} \cr & = 270\,{m^3} \cr} $$

43.

The volume of the largest possible cube that can be inscribed in a hollow spherical ball of radius r cm is :

• (A) $$\frac{2}{\sqrt{3}}{r}^2$$
• (B) $$\frac{4}{3}{r}^2$$
• (C) $$\frac{8}{3\sqrt{3}}{r}^3$$
• (D) $$\frac{1}{3\sqrt{3}}{r}^3$$

Show Answer

 Clearly, the diagonal of the largest possible cube will be equal to the diameter of the sphere Let the edge of the cube be a $$\eqalign{ & \sqrt 3 a = 2r \cr & \Rightarrow a = \frac{2}{{\sqrt 3 }}r \cr} $$ Volume : $$\eqalign{ & = {a^3} \cr & = {\left( {\frac{2}{{\sqrt 3 }}r} \right)^3} \cr & = \frac{8}{{3\sqrt 3 }}{r^3} \cr} $$

44.

Given that 1 cu. cm of marble weights 25 gms, the weight of a marble block 28 cm in width and 5 cm thick is 112 kg. The length of the block is :

• (A) 26.5 cm
• (B) 32 cm
• (C) 36 cm
• (D) 37.5 cm

Show Answer

 Let length = x cm Then, $$\eqalign{ & x \times 28 \times 5 \times \frac{{25}}{{1000}} = 112 \cr & \Rightarrow x = \left( {112 \times \frac{{1000}}{{25}} \times \frac{1}{{28}} \times \frac{1}{5}} \right) \cr & \Rightarrow x = 32\,cm \cr} $$

45.

A hollow spherical metallic ball has an external diameter 6 cm and is $$\frac{1}{2}$$ cm thick. The volume of metal used in the ball is :

• (A) $$37\frac{2}{3}\text{ c}{\text{m}}^3$$
• (B) $$40\frac{2}{3}\text{ c}{\text{m}}^3$$
• (C) $$41\frac{2}{3}\text{ c}{\text{m}}^3$$
• (D) $$47\frac{2}{3}\text{ c}{\text{m}}^3$$

Show Answer

 External radius = 3 cm Internal radius = (3 - 0.5) cm = 2.5 cm Volume of the metal : $$\eqalign{ & = \left[ {\frac{4}{3} \times \frac{{22}}{7} \times \left\{ {{{\left( 3 \right)}^3} - {{\left( {2.5} \right)}^3}} \right\}} \right]{\text{ c}}{{\text{m}}^3} \cr & = \left( {\frac{4}{3} \times \frac{{22}}{7} \times \frac{{91}}{8}} \right){\text{ c}}{{\text{m}}^3} \cr & = \left( {\frac{{143}}{3}} \right){\text{ c}}{{\text{m}}^3} \cr & = 47\frac{2}{3}{\text{ c}}{{\text{m}}^3} \cr} $$

46.

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 cm

Show Answer

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

47.

A swimming bath is 24 m long and 15 m broad. When a number of men dive into the bath, the height of the water rises by 1 cm. If the average amount of water displaced by one of the men be 0.1 cu.m, how many men are there in the bath ?

• (A) 32
• (B) 36
• (C) 42
• (D) 46

Show Answer

 Volume of water displaced : $$\eqalign{ & = \left( {24 \times 15 \times \frac{1}{{100}}} \right){m^3} \cr & = \frac{{18}}{5}{m^3} \cr} $$ Volume of water displaced by 1 man = 0.1 m3 ∴ Number of men : $$\eqalign{ & = \left( {\frac{{\frac{{18}}{5}}}{{0.1}}} \right) \cr & = \left( {\frac{{18}}{5} \times 10} \right) \cr & = 36 \cr} $$

48.

A bucket is in the from of a frustum of a cone and holds 28.490 litres of water. The radii of the top and bottom are 28 cm and 21 cm respectively. Find the height of the bucket.

• (A) 15 cm
• (B) 20 cm
• (C) 25 cm
• (D) 30 cm

Show Answer

 Volume of bucket : = 28.490 litres = (28.490 ×1000) cm3 = 28490 cm3 Let the height of the bucket be h cm We have : r = 21 cm. and R = 28 cm $$\therefore \frac{\pi }{3}h\left[ {{{\left( {28} \right)}^2} + {{\left( {21} \right)}^2} + 28 \times 21} \right]$$     $$ = 28490$$ $$ \Rightarrow h\left( {784 + 441 + 588} \right) = $$     $$\frac{{28490 \times 21}}{{22}}$$ $$\eqalign{ & \Rightarrow 1813h = 27195 \cr & \Rightarrow h = \frac{{27195}}{{1813}} \cr & \Rightarrow h = 15\,cm \cr} $$

49.

A patient in a hospital is given soup daily in a cylindrical bowl of diameter 7 cm. If the bowl is filled with soup to a height of 4 cm, how much soup the hospital has to prepare daily to serve 250 patients ?

• (A) 38L
• (B) 40L
• (C) 39.5L
• (D) 38.5L

Show Answer

 Diameter of bowl = 7 cm ∴ Radius of bowl = $$\frac{2}{7}$$ cm Height = 4 cm ∴ Volume of cylindrical bowl : $$\eqalign{ & = \pi {r^2}h \cr & = \frac{{22}}{7} \times \frac{7}{2} \times \frac{7}{2} \times 4 \cr & = 154\,cu.cm \cr} $$ Hence, volume of soup for 250 patients : $$\eqalign{ & = 154 \times 250 \cr & = 38500{\text{ c}}{{\text{m}}^3} \cr & = 38.5{\text{L}} \cr} $$

50.

The dimensions of a certain machine are 48" × 30" × 52". If the size of the machine is increased proportionately until the sum of its dimensions equal 156", what will be the increase in the shortest side ?

• (A) 4"
• (B) 6"
• (C) 8"
• (D) 9"

Show Answer

 Sum of original dimension : $$\eqalign{ & = 48 + 30 + 52 \cr & = 130 \cr} $$ Increase in sum : $$\eqalign{ & = 156 - 130 \cr & = 26 \cr} $$ Since the dimensions have been increased proportionately, So increase in shortest side : $$ = \left( {26 \times \frac{{30}}{{130}}} \right)$$   " $$ = 6$$ "