Math Ncert Exemplar 2019 Solutions for Class 9 Maths Chapter 13 Surface Area And Volumes are provided here with simple step-by-step explanations. These solutions for Surface Area And Volumes are extremely popular among Class 9 students for Maths Surface Area And Volumes Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Math Ncert Exemplar 2019 Book of Class 9 Maths Chapter 13 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Math Ncert Exemplar 2019 Solutions. All Math Ncert Exemplar 2019 Solutions for class Class 9 Maths are prepared by experts and are 100% accurate.

Question 1:

Write the correct answer in each of the following:
The radius of a sphere is 2r, then its volume will be
(A) $\frac{4}{3}\mathrm{\pi }{r}^{3}$

(B) 4πr3

(C)$\frac{8\mathrm{\pi }{r}^{3}}{3}$

(D) $\frac{32}{3}\mathrm{\pi }{r}^{3}$

Volume of sphere

Hence, the correct answer is option D.

Question 2:

Write the correct answer in each of the following:
The total surface area of a cube is 96 cm2. The volume of the cube is:
(A) 8 cm3
(B) 512 cm3
(C) 64 cm3
(D) 27 cm3

Surface area of cube = 6 × (Side)2
⇒ 96 = 6 × (Side)2
⇒ (Side)2 = 16
⇒ Side = 4 cm

Now, volume of cube = (Side)3
= (4)3
= 64 cm3

Hence, the correct answer is option C.

Question 3:

Write the correct answer in each of the following:
A cone is 8.4 cm high and the radius of its base is 2.1 cm. It is melted and recast into a sphere. The radius of the sphere is :
(A) 4.2 cm
(B) 2.1 cm
(C) 2.4 cm
(D) 1.6 cm

Given: height of cone = 8.4 cm
radius of cone = 2.1 cm
Volume of cone

Since, cone is melted and recast into sphere,
Volume of sphere = Volume of cone

Hence, the correct answer is option B.

Question 4:

Write the correct answer in each of the following:
In a cylinder, radius is doubled and height is halved, curved surface area will be
(A) halved
(B) doubled
(C) same
(D) four times

Let radius be 'r' and height be 'h'
∴ Original Curved Surface Area = $2\pi rh$

Hence, the correct answer is option C.

Question 5:

Write the correct answer in each of the following:
The total surface area of a cone whose radius is $\frac{r}{2}$ and slant height 2l is
(A) 2πr (l + r)
(B) $\mathrm{\pi }r\left(l+\frac{r}{4}\right)$
(C) πr (l + r)
(D) 2πrl

Hence, the correct answer is option B.

Question 6:

Write the correct answer in each of the following:
The radii of two cylinders are in the ratio of 2:3 and their heights are in the ratio of 5 : 3. The ratio of their volumes is:
(A) 10 : 17
(B) 20 : 27
(C) 17 : 27
(D) 20 : 37

Let two cylinders of radius r1 and r2 and height h1 and h2.
Ratio of radius = r: r= 2 : 3
Ratio of height = hh2 = 5 : 3
Ratio of Volumes

Hence, the correct answer is option B.

Question 7:

Write the correct answer in each of the following:
The lateral surface area of a cube is 256 m2. The volume of the cube is
(A) 512 m3
(B) 64 m3
(C) 216 m3
(D) 256 m3

Lateral Surface Area of Cube = 4 × (Side)2
Given: Lateral Surface Area = 256
∴ 256 = 4 × (Side)2
⇒ (Side)2 = 64
⇒ Side = 8 m
Now, Volume of cube = (Side)3
= (8)3
= 512 m3
Hence, volume of cube is 512 m3

Hence, the correct answer is option A.

Question 8:

Write the correct answer in each of the following:
The number of planks of dimensions (4 m × 50 cm × 20 cm) that can be stored in a pit which is 16 m long, 12 m wide and 4 m deep is
(A) 1900
(B) 1920
(C) 1800
(D) 1840

Hence, the correct answer is option B.

Question 9:

Write the correct answer in each of the following:
The length of the longest pole that can be put in a room of dimensions (10 m × 10 m × 5 m) is
(A) 15 m
(B) 16 m
(C) 10 m
(D) 12 m

Given: Dimension of a room l = 10 m, b = 10 m, h = 5 m.
∴ Length of longest pole = Diagonal of cuboid

Hence, the correct answer is option A.

Question 10:

Write the correct answer in each of the following:
The radius of a hemispherical balloon increases from 6 cm to 12 cm as air is being pumped into it. The ratios of the surface areas of the balloon in the two cases is
(A) 1 : 4
(B) 1 : 3
(C) 2 : 3
(D) 2 : 1

Let Initial radius of hemispherical balloon = r1 = 6 cm
Final radius of hemispherical balloon = r2 = 12 cm
∴ Ratio of Surface Area of balloon $=\frac{3\pi {r}_{1}^{2}}{3\pi {r}_{2}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{{r}_{1}^{2}}{{r}_{2}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{{\left(6\right)}^{2}}{{\left(12\right)}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{1}{4}$

Hence, the correct answer is option A.

Question 1:

Write True or False and justify your answer in the given question :
The volume of a sphere is equal to two-third of the volume of a cylinder whose height and diameter are equal to the diameter of the sphere.

True
Let radius of sphere = r
According to question,
height and diameter of cylinder = diameter of sphere
So, radius of cylinder = r, height of cylinder = 2r
Now,

Hence, given statement is true.

Question 2:

Write True or False and justify your answer in the given question :
If the radius of a right circular cone is halved and height is doubled, the volume will remain unchanged.

False
Let original radius of cone = r
and height of cone = h
Volume of cone = $\frac{1}{3}\mathrm{\pi }{r}^{2}h$
Now, radius of cone is halved and height is doubled then.

$\begin{array}{rcl}V& =& \frac{1}{3}\mathrm{\pi }{\left(\frac{r}{2}\right)}^{2}×2h\\ & =& \frac{1}{3}\mathrm{\pi }\frac{{r}^{2}}{4}×2h\\ & =& \frac{1}{2}\left(\frac{1}{3}\mathrm{\pi }{r}^{2}h\right)\\ & & \end{array}$

Hence, new volume = half of original volume.

Question 3:

Write True or False and justify your answer in the given question :
In a right circular cone, height, radius and slant height do not always be sides of a right triangle.

False
Let us consider a right circular cone with height = h, radius = r, slant height = l.

Using Phythagoras theorem,

Thus, height, radius and slant height of cone can always be sides of right triangle.

Here, the given statement is false.

Question 4:

Write True or False and justify your answer in the given question :
If the radius of a cylinder is doubled and its curved surface area is not changed, the height must be halved.

True,
Let radius and height of cylinder be 'r' and 'h'
Then, curved surface area of cylinder = $2\mathrm{\pi }rh$   ......(1)
Now, according to question,
Radius is doubled and curved surface area does not change.

Hence, given statement is true.

Question 5:

Write True or False and justify your answer in the given question :
The volume of the largest right circular cone that can be fitted in a cube whose edge is 2r equals to the volume of a hemisphere of radius r.

True,
Here, Diameter of cone = 2(given)
Also, height of cone = Edge of cube
= 2

= Volume of hemisphere of radius r

Here given statement is true.

Question 6:

Write True or False and justify your answer in the given question :
A cylinder and a right circular cone are having the same base and same height. The volume of the cylinder is three times the volume of the cone.

True,
Let radius of base cylinder and cone be r and height be h.
Volume of cylinder = $\mathrm{\pi }{r}^{2}h$              ..........(1)
Volume of cone = $\frac{1}{3}\mathrm{\pi }{r}^{2}h$               ...........(2)
From (1) and (2)
Volume of cylinder = 3 × Volume of cone

Hence, given statement is true.

Question 7:

Write True or False and justify your answer in the given question :
A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is 1 : 2 : 3.

True,

Now,
Volume of cone : Volume of hemisphere : Volume of cylinder

Hence, given statement is true.

Question 8:

Write True or False and justify your answer in the given question :
If the length of the diagonal of a cube is  $6\sqrt{3}$ cm, then the length of the edge of the cube is 3 cm.

False
Diagonal of Cube of side a cm = $a\sqrt{3}$cm.

Hence, given statement is false.

Question 9:

Write True or False and justify your answer in the given question :
If a sphere is inscribed in a cube, then the ratio of the volume of the cube to the volume of the sphere will be 6 : π.

True,
Let radius of Sphere = r
Since, sphere is inscribed in Cube
Therefore, length of edge of Cube = 2r.
∴ Volume of Cube : Volume of Sphere

Here, given statement is true.

Question 10:

Write True or False and justify your answer in the given question :
If the radius of a cylinder is doubled and height is halved, the volume will be doubled.

True,
Let radius of height of Cylinder be r and h.

Hence, given statement is true.

Question 1:

Metal spheres, each of radius 2 cm, are packed into a rectangular box of internal dimensions 16 cm × 8 cm × 8 cm. When 16 spheres are packed the box is filled with preservative liquid. Find the volume of this liquid. Give your answer to the nearest integer. [Use π = 3.14]

Volume of rectangular box = length × breadth × height

Volume of single sphere

Volume of 16 spheres = 33.5238 × 16
= 535.8933 cm3
∴ Volume of liquid = Volume of box – Volume of 16 spheres

Hence, Volume of liquid will be 488 cm3.

Question 2:

A storage tank is in the form of a cube. When it is full of water, the volume of water is 15.625 m3. If the present depth of water is 1.3 m, find the volume of water already used from the tank.

Volume of storage tank = 15.625 cm3
a3 = 15.625 (∵ Volume of Cube = a3)
a = 2.5 m.
Now, Depth of Water = 1.3 m (given)
⇒ Used water is of height = 1.2 m
So, Volume of water used = 2.5 × 2.5 × 1.2
= 7.5 m3

Question 3:

Find the amount of water displaced by a solid spherical ball of diameter 4.2 cm, when it is completely immersed in water.

We know,
Amount of water displaced by a solid spherical ball, when it is completely immersed is equal to volume of sphere.
∴ Volume of water displaced = Volume of sphere

Question 4:

How many square metres of canvas is required for a conical tent whose height is 3.5 m and the radius of the base is 12 m?

Question 5:

Two solid spheres made of the same metal have weights 5920 g and 740 g, respectively. Determine the radius of the larger sphere, if the diameter of the smaller one is 5 cm.

Let Mass and Volume of sphere 1 be M1 and V1 and Mass and Volume of sphere 2 be M2 and V2.

Since, Mass is directly proportional to volume for same metal.

$\therefore \frac{{\mathrm{M}}_{1}}{{\mathrm{M}}_{2}}=\frac{{\mathrm{V}}_{1}}{{\mathrm{V}}_{2}}$

Also, Volume of sphere is proportional to R3.

Hence, radius of larger sphere is 5 cm.

Question 6:

A school provides milk to the students daily in a cylindrical glasses of diameter 7 cm. If the glass is filled with milk upto an height of 12 cm, find how many litres of milk is needed to serve 1600 students.

Volume of milk in 1 glass

For 1600 students, Milk needed is
= 1600 × 462
= 739200 cm3

Question 7:

A cylindrical roller 2.5 m in length, 1.75 m in radius when rolled on a road was found to cover the area of 5500 m2. How many revolutions did it make?

Given: height of cylinder = 2.5 m
radius of cylinder = 1.75 m
Curved Surface Area of cylinder
Now,

Number of revolutions

Hence, it makes 200 revolutions.

Question 8:

A small village, having a population of 5000, requires 75 litres of water per head per day. The village has got an overhead tank of measurement 40 m × 25 m × 15 m. For how many days will the water of this tank last?

Volume of Tank

Water to population per day

∴ Water Tank last for

Question 9:

Now, let x be the smaller Spherical laddoo

Hence, the number of smaller Spherical laddoo will be 8.

Question 10:

A right triangle with sides 6 cm, 8 cm and 10 cm is revolved about the side 8 cm. Find the volume and the curved surface of the solid so formed.

We know, when a right triangle is revolved about side 8 cm, a cone is formed having radius = 6 cm, height = 8 cm, slant height = 10 cm.

∴ Volume of cone
Also,
Curved Surface Area

Question 1:

A cylindrical tube opened at both the ends is made of iron sheet which is 2 cm thick. If the outer diameter is 16 cm and its length is 100 cm, find how many cubic centimeters of iron has been used in making the tube?

Now, Inner diameter = Outside – 2 × thickness of iron sheet
= 16 – (2 × 2)
= 12 cm

⇒ Inner radius = 6 cm
Thus, Volume of inner cylinder

Volume of iron used = Volume of Cylinder – Volume of hollow space
= 20,096 – 11,304
= 8792 cm3

Question 2:

A semi-circular sheet of metal of diameter 28 cm is bent to form an open conical cup. Find the capacity of the cup.

Let r be the radius and h be the height of conical cup.
Then,
Circumference of base of conical cup = Circumference of semi-circular sheet

Now, capacity of conical cup

Question 3:

A cloth having an area of 165 m2 is shaped into the form of a conical tent of radius 5 m
(i) How many students can sit in the tent if a student, on an average, occupies on the ground?
(ii) Find the volume of the cone.

Let the slant height of conical tent = l m
Radius of base of conical tent = 5 m
(i) Area of Circular base

Now, number of students

(ii) Curved Surface Area of tent = $\mathrm{\pi }rl$

Question 4:

The water for a factory is stored in a hemispherical tank whose internal diameter is 14 m. The tank contains 50 kilolitres of water. Water is pumped into the tank to fill to its capacity. Calculate the volume of water pumped into the tank.

Radius of hemispherical tent = 7 m
Volume of water pumped = Volume of hemispherical tank – 50 kl

Hence, Volume of water pumped is 668.67 m3.

Question 5:

The volumes of the two spheres are in the ratio 64 : 27. Find the ratio of their surface areas.

Ratio of Volume of two Spheres = 64 : 27

i.e.
Let r1 and r2 be the radius of two Spheres
$\therefore \frac{\frac{4}{3}\mathrm{\pi }{r}_{1}^{3}}{\frac{4}{3}\mathrm{\pi }{r}_{2}^{3}}=\frac{64}{27}\phantom{\rule{0ex}{0ex}}⇒\frac{{r}_{1}^{3}}{{r}_{2}^{3}}=\frac{64}{27}\phantom{\rule{0ex}{0ex}}⇒\frac{{r}_{1}}{{r}_{2}}=\frac{4}{3}$

Now, ratio of area of two spheres,

Hence, ratio of area of two Spheres is 16 : 9.

Question 6:

A cube of side 4 cm contains a sphere touching its sides. Find the volume of the gap in between.

Since, the Sphere is touching the sides of Cube
∴ diameter of sphere = Length of Cube side
= 4 m
⇒ radius of Sphere = 2 m
Now,
Volume of gap between Cube and Sphere = (Volume of Cube) – (Volume of Sphere)

Hence, volume of gap between Cube and Sphere is 30.48 cm3.

Question 7:

A sphere and a right circular cylinder of the same radius have equal volumes. By what percentage does the diameter of the cylinder exceed its height?

Let radius of Sphere and Cylinder be r and height of Cylinder be h.
Now,
Volume of Sphere = Volume of Cylinder

Hence, diameter of cylinder exceeds height by 50%.

Question 8:

30 circular plates, each of radius 14 cm and thickness 3 cm are placed one above the another to form a cylindrical solid. Find :
(i) the total surface area
(ii) volume of the cylinder so formed.

Given: radius of Circular Plate = 14 cm
Thickness of Circular Plate = 3 cm
So, height of Cylinder = Thickness of 30 Circular Plates
= 30 × 3
= 90 cm.

(i) Total Surface Area of Solid Cylinder so formed

(ii) Volume of Cylinder so formed

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