**Mensuration Exercise 15.1 – Questions:**

**The height of a right circular cylinder is 14 cm and the radius of its base is 2 cm. Find its (i) CSA (ii) TSA****An iron pipe 0 cm long has external radius equal to 12.5 cm and external radius equal to 11.5 cm. Find the TSA of the pipe.****The radii of two circular cylinders are in the ratio 2:# and the ratio of their curved surface areas is 5:6. Find the ratio of their heights.****The inner diameter of a circular well is 2.8 cm. It is 10 m deep. Find its inner curved surface area. Also find the cost of plastering this curved surface at the rate of Rs. 42 per m**^{2}?**Craft teacher of the school taught the students to prepare cylindrical pen holders out of card board. In a class of strength 42, if each child prepared a pen holder of radius 5 cm and height 14 cm, how much cardboard was consumed?****The diameter of a garden roller is 1.4 m and 2 m long. How much area will it cover in 5 revolutions?****Find the volume of right circular cylinder whose radius is 10.5 cm and height is 16 cm.****The inner diameter of a cylinder wooden pipe is 24 cm and its outer diameter is 28 cm. The pipe is 35 cm long. Find the mass of the pipe if 1cm**^{3}of wood has a mass of 0.6 gm.**Two circular cylinder of equal volumes have their heights in the ratio 1:2. Find the ratio of their radii.****A rectangular sheet of paper, 44 cm x 20 cm is rolled along its length to form a cylinder. Find the volume of the cylinder so formed.**

**Mensuration Exercise 15.1 – Solutions:**

**The height of a right circular cylinder is 14 cm and the radius of its base is 2 cm. Find its (i) CSA (ii) TSA**

Solution:

Given:

Height of a right circular cylinder, h = 14 cm.

Radius = 2 cm

(i) CSA of cylinder = 2πrh

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

= 176 cm^{2}

(ii) TSA of cylinder = 2πr(h + r)

= 2 x ^{22}/_{7} x 2 x (14 + 2)

= 201.14 cm^{2}

**An iron pipe 20 cm long has external radius equal to 12.5 cm and external radius equal to 11.5 cm. Find the TSA of the pipe.**

Solution:

Given:

h = 20 cm

External radius = R = 12.5 cm

Inner radius = r = 11.5 cm

(a) Inner CSA = S_{1} = 2πrh = 2 x ^{22}/_{7} x 11.5 x 20 = 1445.71 cm^{2}

(b) Outer CSA = S_{2} = 2πRh = 2 x ^{22}/_{7} x 12.5 x 20 = 1571.43 cm^{2}

(c) TSA = S_{1} + S_{2} + area of two bases

= S_{1} + S_{2} + 2(πR^{2} – πr^{2})

= 1445.71 + 1571.43 + 2[^{22}/_{7}(_{}12.5)^{2} – ^{22}/_{7}(11.5)^{2}]

= 3167.98 cm^{2}

**The radii of two circular cylinders are in the ratio 2:3 and the ratio of their curved surface areas is 5:6. Find the ratio of their heights.**

Solution:

Let the radii of 2 cylinders be 2r and 3r respectively and their curved surface areas be 5cm^{2} and 6cm^{2} respectively.

We know, Curved surface area S = 2πrh

Then, h = ^{S}/_{2πr}

Let h_{1} be the height of the cylinder of radii 2r and curved surface area 5 cm^{2} = ^{5}/_{2πx2r } = ^{5}/_{4πr}

Let h_{2} be the height of the cylinder of radii 3r and curved surface area 6 cm^{2} = ^{6}/_{2πx3r } = ^{1}/_{πr}

Therefore, h_{1}/h_{2} = (^{5}/_{4πr})/(^{1}/_{πr}) = ^{5}/_{4}

h_{1}: h_{2} = 5: 4

**The inner diameter of a circular well is 2.8 cm. It is 10 m deep. Find its inner curved surface area. Also find the cost of plastering this curved surface at the rate of Rs. 42 per m**^{2}?

Solution:

Inner diameter of a circular well = 2.8 cm r = ^{2.8}/_{2} = 1.4 cm and h = 10 m

Inner curved surface area = 2πrh = 2 x ^{22}/_{7} x 1.4 x 10 = 88 cm^{2}

The cost of plastering for curved surface area 1m^{2} is 42. Then, the cost of plastering 88cm^{2} =Rs. 3696

** **

**Craft teacher of the school taught the students to prepare cylindrical pen holders out of card board. In a class of strength 42, if each child prepared a pen holder of radius 5 cm and height 14 cm, how much cardboard was consumed?**

Solution:

No. of students = 42

radius of the pen holder, r = 5 cm

height of the pen holder, h = 14 cm

The card board consumed for 1 pen holder = curved surface area + area of the base = 2πrh + πr^{2}

= 2 x ^{22}/_{7} x 5 x 14 + ^{22}/_{7} x 5^{2}

= 440 +78.57

= 518.57 cm^{2}

Card board consumed to make 1 pen holder is 518.57cm^{2}. Then the card board consumed to make 42 pen holder = 42×518.57 = 21780 cm^{2}

^{ }

**The diameter of a garden roller is 1.4 m and 2 m long. How much area will it cover in 5 revolutions?**

Solution:

The diameter of the garden roller, d = 1.4 m

Then, radius, r = ^{1.4}/_{2} = 0.7m

Height of the garden roller, h = 2m

Curved surface area = 2πrh

= 2 x ^{22}/_{7} x 0.7 x 2

= 8.8 m^{2}

No. of revolutions garden roller takes is 5. Then the area covered in 5 revolutions = 8.8 x 5 = 44m^{2}

** **

**Find the volume of right circular cylinder whose radius is 10.5 cm and height is 16 cm.**

Solution:

Radius of the right circular cylinder, r = 10.5 cm

Height of the right circular cylinder, h = 16 cm

Volume of a cylinder = πr^{2}h

= ^{22}/_{7} x 10.5^{2} x 16

= 5544 cm^{3}

**The inner diameter of a cylinder wooden pipe is 24 cm and its outer diameter is 28 cm. The pipe is 35 cm long. Find the mass of the pipe if 1cm**^{3}of wood has a mass of 0.6 gm.

Solution:

Inner diameter of a cylinder wooden pipe, d = 24 cm, r = ^{24}/_{2} = 12 cm

Outer diameter of a cylinder wooden pipe, D = 28 cm, d = ^{28}/_{2} = 14 cm

Height of a cylinder wooden pipe, h = 35 cm

Volume of the of a cylinder wooden pipe, V = πR^{2}h – πr^{2}h

= ^{22}/_{7} x 35x(14^{2} – 12^{2})

= 5720 cm^{3}

The mass of the pipe if 1cm^{3} of wood has a mass of 0.6 gm = 5720 x 0.6 = 3432 gm.

**Two circular cylinder of equal volumes have their heights in the ratio 1:2. Find the ratio of their radii.**

Solution:

Ratio of heights of two circular cylinders of equal volumes = 1:2

Therefore, h_{1} = 1 and h_{2} = 2

Volume of the circular cylinder = πr^{2}h

Given, V_{1} = V_{2} = V

Radius of the circular cylinder of h_{1} = 1, πr_{1}^{2}h = V, r_{1}^{2} = ^{V}/_{πh1}

Radius of the circular cylinder of ^{ }h_{2} = 2, πr_{2}^{2}h = V, r_{2}^{2} = ^{V}/_{πh2}

Ratio of their radii = r_{1}^{2}/r_{2}^{2 }= (^{V}/_{πh1})/ (^{V}/_{πh2}) = ^{h2}/_{h1} = ^{2}/_{1}

^{r1}/_{r2} = √(^{2}/_{1}) = ^{√2}/_{1}

Therefore, ratio of their radii = √2:1

**A rectangular sheet of paper, 44 cm x 20 cm is rolled along its length to form a cylinder. Find the volume of the cylinder so formed.**

Solution:

Area of rectangular sheet = Curved surface area of the cylinder =

⇒ l x b = 2πr x h

⇒ 44cm x 20cm = l x b

Thus, h = 20 cm and 2πr = 44 cm

⇒ 2πr = 44 cm

⇒ πr = 22

⇒ r = ^{22×7}/_{22} = 7 cm

Volume of the cylinder = πr^{2}h

= ^{22}/_{7} x 7^{2} x 20

= 3080 cm^{3}