RS Aggarwal solution class 8 chapter 20 Volume and Surface Area of Solid Test Paper 20

Test Paper 20

Page-231

Question 1:

Find the volume of a cube whose total surface area is 384 cm².

Sol :

Total surface area$=6a^2$
$\Rightarrow 6a^2=384$
$\Rightarrow a=\sqrt{\frac{384}{6}}=8 cm$
$\therefore \mathrm{Volume}=a^3=512 c{m}^3$

Question 2:

How many soap cakes each measuring 7 cm × 5 cm × 2.5 cm can be placed in a box of size 56 cm × 40 cm × 25 cm?

Sol :

Volume of a soap cake $=7\times 5\times 2.5=87.5 c{m}^3$
Volume of the box$=56\times 40\times 25=56000 c{m}^3$

No. of soap cakes$=\frac{56000}{87.5}=640 units$

∴ 640 cakes of soap can be placed in a box of the given size.

Question 3:

The radius and height of a cylinder are in the ratio 5 : 7 and its volume is 550 cm³. Find its radius and height.

Sol :

$$\begin{gathered} \frac{\mathrm{Radius}}{\mathrm{height}}=\frac{r}{h}=\frac{5}{7} \\ \Rightarrow r=\frac{5}{7}h \end{gathered}$$
Now, volume$={\mathrm{\pi r}}^2\mathrm{h}=\frac{22}{7}\times \frac{5}{7}\mathrm{h}\times \frac{5}{7}\mathrm{h}\times \mathrm{h}=550 c{m}^3$
$$\begin{gathered} \therefore h=\sqrt[3]{\frac{550\times 7\times 7\times 7}{22\times 5\times 5}}=7 cm \\ \mathrm{Also}, r=\frac{5}{7}h=5 cm \end{gathered}$$

Question 4:

Find the number of coins, 1.5 cm in diameter and 0.2 cm thick, to be melted to form a right circular cylinder with a height of 10 cm and a diameter of 4.5 cm.

Sol :

Volume of the coin$={\mathrm{\pi r}}^2\mathrm{h}=\frac{22}{7}\times 0.75\times 0.75\times 0.2$
Volume of the cylinder $={\mathrm{\pi r}}^2\mathrm{h}=\frac{22}{7}\times 2.25\times 2.25\times 10$
No. of coins$=\frac{\mathrm{volume} \mathrm{of} \mathrm{cylinder}}{}$

$\frac{\mathrm{volume} \mathrm{of} \mathrm{coin}}{}=\frac{2.25\times 2.25\times 10}{0.75\times 0.75\times 0.2}= 450 \mathrm{coins}$

∴ 450 coins must be melted to form the required cylinder.

Question 5:

Find the surface area of a chalk box, whose length, breadth and height are 18 cm, 10 cm and 8 cm respectively.

Answer 5:

Length = 18 cm
Breadth = 10 cm
Height = 8 cm
∴ Total surface area $=2\left(lb+lh+bh\right)$

$=2\left(18\times 10+18\times 8+10\times 8\right)$

$=2\left(180+144+80\right)=808 c{m}^2$  

Question 6:

The curved surface area of a cylindrical pillar is 264 m² and its volume is 924 m³. Find the diameter and height of the pillar.

Sol :

Curved surface area $=2\mathrm{\pi rh}=264 {\mathrm{m}}^2$
$\therefore r=\frac{264}{2\mathrm{\pi h}}=\frac{132}{\mathrm{\pi h}}m$

Volume $={\mathrm{\pi r}}^2\mathrm{h}=\mathrm{\pi }\times \frac{132}{\mathrm{\pi h}}\times \frac{132}{\mathrm{\pi h}}\times \mathrm{h}=924 {\mathrm{m}}^3$
$\therefore h=\frac{132\times 132\times 7}{22\times 924}=6 m$

Now, $r=\frac{132}{\mathrm{\pi h}}=\frac{132\times 7}{22\times 6}=7m$

i.e., diameter of the pillar, $d=7\times 2=14 m$

Question 7:

Mark (✓) against the correct answer:
The circumference of the circular base of a cylinder is 44 cm and its height is 15 cm. The volume of the cylinder is
(a) 1155 cm³
(b) 2310 cm³
(c) 770 cm³
(d) 1540 cm³

Sol :

(b) 2310 cm³

Height = 15 cm
Circumference$=2\mathrm{\pi r}=44 \mathrm{cm}$
$\therefore r=\frac{44\times 7}{2\times 22}=7 cm$
∴ Volume$={\mathrm{\pi r}}^2\mathrm{h}=\frac{22}{7}\times 7\times 7\times 15=2310 {\mathrm{cm}}^3$

Question 8:

Mark (✓) against the correct answer:
the area of the base of a circular cylinder is 35 cm² and its height is 8 cm. The volume of the cylinder is
(a) 140 cm³
(b) 280 cm³
(c) 420 cm³
(d) 210 cm³

Sol :

(b) 280 cm³
Area = 35 cm²
Height = 8 cm
$\therefore \mathrm{Volume} = \mathrm{base} \mathrm{area} \times \mathrm{height} = 35\times 8=280 {\mathrm{cm}}^3$

Question 9:

Mark (✓) against the correct answer:
A cuboid having dimensions 16 m × 11 m × 8 m is melted to form a cylinder of radius 4 m. What is the height of the cylinder?
(a) 28 m
(b) 14 m
(c) 21 m
(d) 32 m

Sol :

(a) 28 m 
Volume of the cuboid$=16\times 11\times 8=1408 {\mathrm{m}}^3$
​Volume of the cylinder $={\mathrm{\pi r}}^2\mathrm{h}=1408 {\mathrm{m}}^3$
$\therefore h=\frac{1408\times 7}{22\times 4\times 4}=28 \mathrm{m}$

Question 10:

Mark (✓) against the correct answer:
The dimensions of a cuboid are 8 m  6 m  4 m. Its lateral surface area is
(a) 210 m²
(b) 105 m²
(c) 160 m²
(d) 240 m²

Sol :

Lateral surface area $=2\left(\left(l+b\right)\times h\right)=2\left(\left(8+6\right)\times 4\right)=2\left(56\right)=112 {\mathrm{m}}^2$

Question 11:

Mark (✓) against the correct answer:
The length, breadth and height of a cuboid are in the ratio 3 : 4 : 6 and its volume is 576 cm³. The whole surface area of the cuboid is
(a) 216 cm²
(b) 324 cm²
(c) 432 cm²
(d) 460 cm²

Sol :

(c) 432 sq cm
Volume$=lbh=3x\times 4x\times 6x=72x^3 =576 c{m}^3$
$\Rightarrow x=\sqrt[3]{\frac{576}{72}}=2$
∴ Total surface area$=2\left(lb+bh+lh\right)$

$=2(3x4x+4x6x+3x6x)$

$=2(48+96+72)=432 {\mathrm{cm}}^2$

Question 12:

Mark (✓) against the correct answer:
The surface area of a cube is 384 cm². Its volume is
(a) 512 cm³
(b) 256 cm³
(c) 384 cm³
(d) 320 cm³

Sol :

(a) 512 cm³
Surface area$=6a^2$
$\Rightarrow 6a^2=384$
$\Rightarrow a=\sqrt{\frac{384}{6}}=\sqrt{64}=8 \mathrm{cm}$
∴ Volume$=a^3=8^3=512 {\mathrm{cm}}^3$

Question 13:

Fill in the blanks.
(i) If l, b, h be the length, breadth and height of a cuboid, then its whole surface area = (.......) sq units.
(ii) If l, b, h be the length, breadth and height of a cuboid, then its lateral surface area = (.......) sq units.
(iii) If each side of a cube is a, then its lateral surface area is ....... sq units.
(iv) If r is the radius of the base and h be the height of a cylinder, then its volume is (.......) cubic units.
(v) If r is the radius of the base and h be the height of a cylinder, then its lateral surface area is (......) sq units.

Sol :

(i) If l, b and h are the length, breadth and height of a cuboid, respectively, then its whole surface area is equal to $2(lb+lh+bh)$ sq units.
(ii) If l, b and h are the length, breadth and height of a cuboid, respectively, then its lateral surface area is equal to $2\left(\right(l+b)\times h)$ sq units.
(iii) If each side of a cube is a, then the lateral surface area is $4a^2$ sq units.
(iv) If r and h are the radius of the base and height of a cylinder, respectively, then its volume is $\mathrm{\pi }{r}^{2}h$ cubic units.
(v) If r and h are the radius of the base and height of a cylinder, then its lateral surface area is $2\mathrm{\pi }rh$ sq units.