## RS Aggarwal Class 9 Ex 13C Chapter 13

**Q.1: ****Find the volume and surface area of a sphere whose radius is (i) 3.5 cm (ii) 4.2 cm (iii) 5 m**

**Solution:**

**(i)** Radius of sphere = 3.5 cm

∴ Volume of the sphere = ^{3}

∴ Surface area of the sphere = 4𝜋r^{2} = ^{2}

**(ii)** Radius of the sphere = 4.2 cm

∴ Volume of the sphere = ^{3}

∴ Surface area of the sphere = 4𝜋r^{2} = ^{2}

**(iii)** Radius of the sphere = 5 m

∴ Volume of the sphere =

= ^{3}

∴ Surface area of the sphere = 4𝜋r^{2}= ^{2}

**Q.2:** **The volume of a sphere is 38808 cm ^{3}. Find its radius and hence its surface area.**

**Solution:**

Volume of the sphere =

^{3})

∴ Surface area of the sphere = 4𝜋r^{2} = ^{2}

**Q.3:** **Find the surface area of a sphere whose volume is 606.375 m ^{3}**

**Solution:**

Volume of the sphere = 606.375 m^{3}

Volume of the sphere =

^{3})

∴ Surface area of the sphere = 4𝜋r^{2} = ^{2}

**Q.4:** **The surface area of a sphere is 394.24 m ^{2}. Find its radius and volume.**

**Solution:**

Let the radius of the sphere be ‘r’ m.

Then, its surface area = 4𝜋r^{2}

∴ 4𝜋r^{2} = 394.24 [given]

∴ Radius of the sphere = 5.6 m

Volume of the sphere =

= ^{3}

∴ Volume of the sphere = 735.91 m^{3}

**Q.5:** **The surface area of a sphere is (576𝜋) cm ^{2}. Find its volume.**

**Solution:**

Surface area of sphere = 4𝜋r^{2}

∴ 4𝜋r^{2 }= 576𝜋 [surface area = 576𝜋 cm^{2}]

∴ Volume of the sphere =

= ^{3}

∴ Volume of the sphere = 2304𝜋 cm^{3}

**Q.6:** **The outer diameter of a spherical shell is 12 cm and its inner diameter is 8 cm. Find the volume of metal contained in the shell. Also, find its outer surface area.**

**Solution:**

Outer diameter of the spherical shell = 12 cm

Hence, radius = 6 cm

Inner diameter of spherical shell = 8 cm

Hence, radius = 4 cm

Now, Volume of the outer shell = ^{3}

And, Volume of the inner shell = ^{3}

Volume of metal contained in the shell = (Volume of outer) – (Volume of inner) = (905.15) -(268.20) cm^{3} = 636.95 cm^{3}

∴ Outer Surface area = 4𝜋r^{2} = ^{2}

**Q.7:** **How many lead shots, each of 3 mm in diameter can be made from a cuboid with dimensions (12 cm × 11 cm × 9 cm)?**

**Solution:**

Here, diameter of the lead shot = 3 mm

Hence, radius =

Now, number of lead shots =

=

∴ Number of lead shots = 84000.

**Q.8:** **How many lead balls, each of radius 1 cm, can be made from a sphere of radius 8 cm?**

**Solution:**

Here, radius of 1 lead ball = 1 cm and radius of sphere = 8 cm

But, number of lead balls =

=

=

=

∴ Number of lead balls = 512

**Q.9:** **A solid sphere of radius 3 cm is melted and then recast into smaller spherical balls, each of diameter 0.6 cm. Find the number of small balls thus obtained.**

**Solution:**

Here, radius of sphere = 3 cm

Diameter of spherical ball = 0.6 cm

Radius of spherical ball = 0.3 cm

Number of balls =

=

∴ Number of small balls obtained = 1000

**Q.10:** **A metallic sphere of radius 10.5 cm is melted and then recast into smaller cones, each of radius 3.5 cm and height 3 cm. How many cones are obtained?**

**Solution:**

Here, radius of sphere = 10.5 cm =

Radius of smaller cone = 3.5 cm =

Now, number of cones =

=

∴ Number of cones obtained = 126

**Q.11:** **How many spheres 12 cm in diameter can be made from a metallic cylinder of diameter 8 cm and height 90 cm?**

**Solution:**

Diameter of the sphere = 12 cm

Hence, radius = 6 cm

Therefore, Volume of the sphere =

= **. . . . . . . . . . (i)**

Diameter of cylinder = 8 cm

Hence, the radius = 4 cm

Height of the cylinder = 90 cm

Therefore, Volume of the outer shell =

=

Number of spheres =

NUmber of spheres =

Solving the above equation, we get, number of spheres = 5.

**Q.12:** **The diameter of a sphere is 6 cm. It is melted and drawn into a wire of diameter 2 mm. Find the length of the wire.**

**Solution:**

Here, diameter of a sphere = 6 cm

Hence, radius = 3 cm

Diameter of wire = 2 mm

Hence, its radius = 1 mm = 0.1 cm

Let the required length of the wire be ‘h’ cm.

Then,

=

Hence, the length of the wire = 36 m

**Q.13:** **The diameter of a copper sph****ere is 18 am. It is melted and drawn into a long wire of uniform cross section. If the length of the wire is 108 m, find its diameter.**

**Solution:**

Here, diameter of sphere = 18 cm

Hence, radius = 9 cm

Length of the wire = 108 m = 10800 cm

Then,

So, the diameter = (2) (0.3) = 0.6 cm.

**Q.14:** **A sphere of diameter 15.6 cm is melted and cast into the right circular cone of height 31.2 cm. Find the diameter of the base of the cone.**

**Solution:**

Here, diameter of sphere = 15.6 cm

∴ Radius of sphere =

And, height of cone = 31.2 cm

Then,

∴ Diameter of cone = (2) (7.8) cm = 15.6 cm.

**Q.15:** **A spherical cannon ball 28 cm in diameter is melted and cast into a right circular cone mould, whose base is 35 cm in diameter. Find the height of the cone.**

**Solution:**

Here, diameter of sphere = 28 cm

∴ radius of sphere =

Diameter of cone = 35 cm

∴ the radius of cone =

∴

∴ Height of the cone = 35.84 cm

**Q.16:** **A spherical ball of radius 3 cm is melted and recast into three spherical balls. The radii of two of these balls are 1.5 cm and 2 cm. Find the radius of the third ball.**

**Solution:**

Let the radius of the third ball be ‘r’ cm

Then,

27 =

i.e.

=

i.e.

r =

∴ Radius of the third ball = 2.5 cm.

**Q.17:** **The radii of two spheres are in the ratio 1 : 2. Find the ratio of their surface areas.**

**Solution:**

Let the radii of two spheres be ‘x’ and ‘2x’ and their respective surface areas be S_{1} and S_{2}

Then,

∴ The ratio of their surface areas = 1: 4.

**Q.18:** **The surface areas of two spheres are in the ratio 1 : 2. Find the ratio of their volumes.**

**Solution:**

Let the radii of two spheres be ‘r’ and ‘R’

Then,

Let V_{1} and V_{2} be the volumes of the respective spheres whose radii are r and R.

∴

∴ The ratio of their volume = 1: 8.

**Q.19:** **A cylindrical tube of radius 12 cm contains water to a depth of 20 cm. A spherical iron ball is dropped into the tub and thus the level of water is raised by 6.75 cm. What is the radius of the ball?**

**Solution:**

Let the radius of the ball be ‘r’ cm and ‘R’ be the radius of the cylindrical tub.

Then,

∴ The Radius of the ball = 9 cm.

**Q.20:** **A cylindrical bucket with base radius 15 cm is filled with water up to a height of 20 cm. A heavy iron spherical ball of radius 9 cm is dropped into the bucket to submerge completely in the water. Find the increase in the level of water.**

**Solution:**

Radius of the cylindrical bucket = 15 cm

Height of the cylindrical bucket = 20 cm

Volume of the water in the bucket = 𝜋 × 15 × 15 × 20 cm^{3}

Radius of the spherical ball = 9 cm

Volume of the spherical ball = ^{3} . . . . . . . . (1)

Increase in the water level = h cm

Volume of the increased water level = 𝜋 × 15 × 15 × h cm^{3} . . . . . . . (2)

Equating (1) and (2), we have,

𝜋 × 15 × 15 × h =

h =

h = 4.32 cm

**Q.21:** **A hemispherical of the lead of radius 9 cm is cast into the right circular cone of height 72 cm. Find the radius of the base of the cone.**

**Solution:**

Radius of hemisphere = 9 cm

Height of cone = 72 cm

Let the radius of the base of cone be ‘r’ cm

Then,

∴ The radius of the base of the cone = 4.5 cm.

**Q.22:** **A hemispherical bowl of internal radius 9 cm contains a liquid. This liquid is to be filled into the cylindrical shaped small bottle of diameter 3 am and height 4 cm. How many bottles are required to empty the bowl?**

**Solution:**

Here, internal radius of hemisphere bowl ® = 9 cm

Diameter of bottle = 3 cm

And , height of bottle = 4 cm

Number of bottles =

=

=

=

=

∴ The number of bottles required = 54

**Q.23:** **A hollow spherical shell is made of a metal of density 4.5 g/cm2. If its internal and external radii are 8 cm and 9 cm respectively, find the weight of the shell.**

**Solution:**

Internal radius (r) = 8 cm

External radius (R) = 9 cm

Density of metal = 4.5 g per cm^{3}

∴ weight of the shell =

=

=

=

∴ weight of the shell = 4.092 kg.