**Mensuration Exercise 15.2 – Questions**

**Find the curved surface area of a cone, if its slant height is 60 cm and the radius of its base is 21 cm.****The radius of a cone is 7 cm and area of curved surface is 176 cm**^{2}. Find its slant height.**The area of the curved surface of a cone is 60πcm**^{2}. If the slant height of the cone is 8 cm, find the radius of the base.**Curved surface area of a cone is 308cm**^{2}and its slant height is 14 cm. Find the radius of the base and total surface area of the cone.**A clown’s cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the sheet required to make 10 such caps.****Find the ratio of the curved surface areas of two cones if their diameters of the bases are equal and slant heights are in the ratio 4:3.****Find the volume of a right circular cone with radius 5 cm and height 7 cm.****Two cones have their heights in the ratio 1:3 and the radii of their bases in the ratio 3:1. Find the ratio of their volumes.****A right angled triangle of which the sides containing the right angle are 6.3 cm and 10 cm in length, is made to turn around on the longer side. Find the volume of the solid thus generated.****A tent is of the shape of a right circular cylinder up to a height of 3m and then becomes a right circular cone with a maximum height of 13.5 m above the ground. Calculate the cost of painting the inner side of the tent at the rate of Rs. 2 per sq. m., if the radius of the base is 14m.**

**Mensuration Exercise 15.2 – Solutions**

**Find the curved surface area of a cone, if its slant height is 60 cm and the radius of its base is 21 cm.**

Solution:

Radius, r = 21 cm

Slant height, l = 60 cm

Curved surface area of the cone = πrl

= ^{22}/_{7} x 21 x 60

= 3960 cm^{2}

**The radius of a cone is 7 cm and area of curved surface is 176 cm**^{2}. Find its slant height.

Solution:

Radius, r = 7 cm

Curved surface area of the cone = 176 cm^{2}.

We have to find the slant height, l.

We know, curved surface area of the cone = πrl.

176 = ^{22}/_{7 }x 7 x *l*

*l *= ^{176 x 7}/_{7 x 21 }

*l* = 8 cm

**The area of the curved surface of a cone is 60πcm**^{2}. If the slant height of the cone is 8 cm, find the radius of the base.

Solution:

Curved surface area of the cone = 60π cm^{2}

Slant height, l = 8 cm

We have to find the radius of the cone. Then,

Curved surface area = πrl

60π = π x r x 8

r = ^{60π}/_{8π} = 7.5 cm

**Curved surface area of a cone is 308cm**^{2}and its slant height is 14 cm. Find the radius of the base and total surface area of the cone.

Solution:

Curved surface area of the cone = 308 cm^{2}

Slant height = 14 cm

To find the radius of the cone, we have,

Curved surface area = πrl

308 = π x r x 14

r = ^{308×7}/_{22×14}

r = 7 cm.

Total surface area of the cone = πr (r + l)

= ^{22}/_{7} x 7(7 + 14)

= ^{22}/_{7} x 7(21)

= 462 cm^{2}

Therefore, radius of the cone is 7 cm and total surface area of the cone is 462 cm^{2}.

**A clown’s cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the sheet required to make 10 such caps.**

Solution:

Radius = 7 cm

Height = 24 cm

We know that, l^{2} = r^{2} + h^{2}

l^{2} = 7^{2} + 24^{2} = 625

l = √625 = 25 cm

Curved surface area = πrl

= ^{22}/_{7} x 7 x 25

= 550 cm^{2}

The sheet required to make 1 cap is 550 cm^{2} .Then the area of the sheet required to make 10 such caps = 550 x 10 = 5500 cm^{2}

**Find the ratio of the curved surface areas of two cones if their diameters of the bases are equal and slant heights are in the ratio 4:3.**

Solution:

Let diameters of the cone be d_{1 }and d_{2}.

Slant heights be l_{1} and l_{2}.

Given, diameters of two cones are equal then their radius are also equal.Their slant heights of the cone 4:3 i.e., l_{1} = 4 and l_{2} = 3

We have to find the curved surface areas of two cones. let its be A_{1} and A_{2}.

We know, curved surface area = πrl

Curved surface area of the cone with A_{1} = πrl_{1} = 4πr

Curved surface area of the cone with A_{2} = πrl_{2} = 3πr

^{A1}/_{A2} = ^{4πr}/_{3πr} = ^{4}/_{3}

Ratio of the curved surface area of two cones = 4:3

**Find the volume of a right circular cone with radius 5 cm and height 7 cm.**

Solution:

Radius = 5 cm

height = 7cm

Volume of the cone = ^{1}/_{3} πr^{2}h

= ^{1}/_{3} x ^{22}/_{7} x 5^{2} x 7

= 183.33 cm^{3}

**Two cones have their heights in the ratio 1:3 and the radii of their bases in the ratio 3:1. Find the ratio of their volumes.**

Solution:

Height of the cones be h_{1} = 1 and h_{2} = 3.

Radii of two cones be r_{1} = 3 and r_{2} = 1

Volume of the cone with h_{1} is 1 and r_{1 }is 3 = ^{1}/_{3} πr^{2}h

= ^{1}/_{3} π x (3)^{2 }x 1

= 3π

Volume of the cone with h_{2} is 3 and r_{2 }is 1 = ^{1}/_{3} πr^{2}h

= ^{1}/_{3} π x (1)^{2 }x 3

= π

Ratio of volumes of two cones = ^{v1}/_{v2} = ^{3π}/_{π} = ^{3}/_{1}

Then, Ratio of volumes of two cones is 3:1

**A right angled triangle of which the sides containing the right angle are 6.3 cm and 10 cm in length, is made to turn around on the longer side. Find the volume of the solid thus generated.**

Solution:

Radius = 6.3 cm

height = 10 cm

Volume = ^{1}/_{3} πr^{2}h

= ^{1}/_{3} x ^{22}/_{7} x (6.3)^{2 }x (10)

= 415.8 cm^{3}

^{ }

**A tent is of the shape of a right circular cylinder up to a height of 3m and then becomes a right circular cone with a maximum height of 13.5 m above the ground. Calculate the cost of painting the inner side of the tent at the rate of Rs. 2 per sq. m., if the radius of the base is 14m.**

Solution:

Total height of the tent = 13.5 m

Height of the cylinder = 3 m

Therefore, height of the cone = total height of the tent – height of the cone

= 13.5 – 3

= 10.5 m

Given, radius = 14 m

We know, slant height of the cone , l^{2} = r^{2} +h^{2}

= 14^{2} + 10.5^{2}

= 196 + 110.25

l^{2} = 306.25

l = 17.5 m

Curved surface area of the cone = πrl

= ^{22}/_{7} x 14 x 17.5

= 770 m^{2}

Curved surface area of the cylinder = 2πrh

= 2 x ^{22}/_{7} x 14 x 10.5

= 264 m^{2}

Total curved surface area = curved surface area of the cone + curved surface area of the cylinder

= 770 + 264

= 1034 m^{2}

Therefore, cost of painting inter side of the tent at the rate of Rs. per sq. m = 2 x 1034 = Rs. 2068