Chapter 19: Surface Area And Volume of a Right Circular Cylinder Exercise – 19.1

Question: 1

Curved surface area of a right circular cylinder is 4.4 m².If the radius of the base of the cylinder is 0.7m.Find its height. (Take π = 3.14)

Solution:

Given that

Radius of the base of the cylinder (r) = 0.7 m

Curved surface area of cylinder (C.S.A) = 4.4 m²

Let the height of the cylinder be h

The curved surface area of a cylinder is given by: 2πrh

2πrh = 4.4 m²

2 * 3.14 * 0.7 * h = 4.4 m²

h = 4.4 ÷ 2 ∗ 3.14 ∗ 0.7

= 1 m

Therefore the height of the cylinder is 1 m.

Question: 2

In a hot water heating system, there is a cylindrical pipe of length 28 m and diameter 5 cm. Find the total radiating surface in the system. (Take π = 3.14)

Solution:

Given that

Height of cylinder = length of cylindrical pipe

= 28 m

Radius(r) of circular end of pipe = 5/2 = 2.5 cm = 0.025 m

Curved surface area of cylindrical pipe = 2πrh = 2 * 3.14 * 0.035 * 30 = 4.4 m²

Therefore the area of radiating surface of the system is 4.4 m² or 44000 cm².

Question: 3

A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the cost of painting the curved surface of the pillar at the rate of Rs 12.50 per m².(Take   π = 3.14)

Solution:

Given that

Height of cylindrical pillar = 3.5 m

Radius(r) of circular end of pillar = 50/2 = 25 cm = 0.25 m

Curved surface area of cylindrical pillar = 2π = 2 * 3.14 * 0.25 * 3.5

= 5.5 m²

The cost of whitewashing 1 m² is Rs 12.50

Cost of whitewashing 5.5m² area = Rs (12.5 * 5.5) = Rs 68.75

Thus the cost of whitewashing the pillar is Rs 68.75

Question: 4

It is required to make a closed cylindrical tank of height 1m and the base diameter of 140 cm from a metal sheet. How many square meters of the sheet are required for the same? (Take π = 3.14).

Solution:

Height of the cylindrical tank (h) = 1 m

Base radius of cylindrical tank (r) = 140/2 = 70 cm = 0.7 m

Area of sheet required =total surface area of tank = 2πrh

= 2 * 3.14 * 0.7(0.7 + 1)

= 4.4 * 1.7 = 7.48 m²

Therefore it will require 7.48 m² of metal sheet.

Question: 5

A solid cylinder has a total surface area of 462 cm². Its curved surface area is one-third of its total surface area. Find the radius and height of the cylinder. (Take π = 3.14).

Solution:

Given that

Curved or lateral surface area = 13 * total surface area

2πrh = 1/3 (2πrh + 2πr²)

4πrh = 2πr²

2h = r

Total surface area = 462 cm²

Curved surface area = 1/3 ∗ 462

2πrh = 154

2 ∗ 3.14 ∗ 2 ∗ h² = 154

h² = 49/4

h = 49/4 cm

= 7/2 cm

Now r = 2h

Therefore r = 2 * 7/2 cm = 7cm

The height and the radius of the cylinder is 7/2 cm and 7 cm respectively.

Question: 6

The total surface area of a hollow cylinder which is open on both the sides is 4620 sq.cm and the area of the base ring is 115.5 sq.cm and height is 7cm. Find the thickness of the cylinder.

Solution:

Let the inner radius of the hollow cylinder be r₁ cm

The outer radius of the hollow cylinder be r₂ cm

Then,

Now solving eq (a)

Now putting the value of (b) in (a) we get

⟹ 2πh(r₁ + r₂) + 231 = 4620

⟹ 2π ∗ 7(r₁ + r₂) = 4389

⟹ π(r₁ + r₂) = 313.5 ... (c)

Now solving eq (b)

Dividing equation (d) by (c) we get

⟹ r₂ − r₁ = 0.3684 cm

Question: 7

Find the ratio between the total surface area of a cylinder to its curved surface area, given that height and radius of the tank are 7.5 m and 3.5 m respectively.

Solution:

Given that,

Radius of the cylinder (r) = 3.5 m

Height of the cylinder (h) = 7.5 m

Total Surface Area of cylinder (T. S. A)

= 2πr(r + h)

Curved surface area of a cylinder (C.S.A)

= 2πrh

Now,

Putting the values in eq (1)

= 22/15

= 22:15

Therefore the ratio is 22:15.

Question: 8

The total surface of a hollow metal cylinder, open at both ends of an external radius of 8cm and height 10 cm is 338 cm². Take r to be the inner radius, obtain an equation in r and use it to obtain the thickness of the metal in the cylinder.

Solution:

Given that

The external radius of the cylinder (R) = 8 cm

Height of the cylinder (h) = 10 cm

The total surface area of the hollow cylinder (T.S.A) = 338π cm²

As we know that,

2πr∗h + 2πR∗h + 2πR∗2 − 2πr² = 338π cm²

⟹ h(r + R)+(R + r)(R − r) = 169

⟹ 10(8 + r) + (8 + r)(8 − r) = 169

⟹ 80 + 10r + 64 − r² = 169

⟹ r² − 10r + 25 = 0

⟹ r = 5

R − r = 8 – 5 cm = 3 cm

Question: 9

A cylindrical vessel, without lid, has to be tin-coated on its both sides. If the radius of the base is 70 cm and its height is 1.4 m, calculate the cost of tin-coating at the rate of Rs 3.50 per 1000 cm².

Solution:

Given that

Radius of the vessel (r) = 70 cm

Height of the vessel (h) = 1.4 m = 140 cm

The area to be tin coated

= 2 * (2πrh + πr²)

= 2πr (2h + r)

= 2 * 3.14 * 70[(2 * 140) + 70]

= 154000 cm²

Question: 10

The inner diameter of a circular well is 3.5 m. It is 10 m deep. Find:

(a) Inner curved surface area

(b) The cost of plastering this curved surface at the rate of Rs 40 per m².

Solution:

The inner diameter of the well = 3.5 m

Inner radius = 3.52 = 1.75 m

Height of the well = 10 m

(a) Inner curved surface area

= 2πrh

= 2 * 3.14 * 1.75 * 10

= 110m²

(b) Cost of painting 1m² area of the well

= Rs 40

Cost of painting 110m² area of the well

= Rs (50*110) = Rs 4400

Question: 11

Find the lateral surface area of a petrol storage tank is 4.2 m in diameter and 4.5 m high. How much steel was actually used, if 1/12^th of the steel actually used was wasted in making the closed tank?

Solution:

It is given that

Diameter of cylinder = 4.2 m

Radius of cylinder = 4.22 m

= 2.1 m

Height of cylinder = 4.5 m

Therefore,

Lateral or Curved surface area = 2πrh

= 2 * 3.14 * 2.1 * 4.5 = 59.4 m²

Total surface area of tank = 2*?*r(r + h)

=2 * (22/7 * 2.1(2.1 + 4.5) m²

= 87.12 m²

Let, A m² steel be actually used in making the tank

Area of iron present in cylinder

= 11/12 Am²

Hence,

11/12A = Total surface area of cylinder

⟹ A = 12/11*Total surface area

⟹ A = (12/11∗ 87.12) m²

= 95.04 m²

Thus, m² steel was actually wasted while constructing a tank.

Question: 12

The students of Vidyalaya were asked to participate in a competition for making and decorating penholders in the shape of a cylinder with a base using cardboard. Each penholder was to be of radius 3 cm and height 10.5 cm. The Vidyalaya was to supply the competitors with cardboard. If there were 35 competitors, how much cardboard was required to be bought for the competition? (Take π = 3.14).

Solution:

It is given that

Radius of the circular part of the penholder (r) = 3 cm

The height of the penholder (h) = 10.5 cm

Surface area of one penholder (S.A)

= Curved surface area of penholder + Area of the circular base of penholder

= 2πrh + πr²

= (2 * 3.14 * 3 * 10.5) + 3.14 * 3²

= 198 + 198/7

= 1584/7 cm²

The total area of cardboard sheet used by one competitor = 1584/7 cm²

The total area of cardboard sheet used by 30 competitors = 1584/7 * 35 cm²

= 7920 cm²

Therefore, the school needs to buy 7920 cm² of cardboard sheet for the competition.

Question: 13

The diameter of roller 1.5 m long is 84 cm.If it takes 100 revolutions to level a playground, find the cost of leveling this ground at the rate of 50 paise per square meter.

Solution:

Given that

Diameter of the roller (d) = 85 cm = 0.84 m

Length of the roller = 1.5 m

Radius of the roller(r) = d/2

= 0.842 = 0.42 m

The total area of the playground covered by the roller in one revolution=covered area of the roller

Curved surface area of the roller = 2πrh

= 2 * 3.14 * 0.42 * 1.5

= 0.12 * 22 * 1.5 m²

Area of the playground = 100 * Area covered by roller in one revolution

= (100 * 0.12 * 22 *1.5) m²

= 396 m²

Now,

Cost of leveling 1 m² = 50p = Rs 0.5

Cost of leveling 396 m² = Rs.396 * 0.5 = Rs.198

Hence, cost of leveling 396 m² is Rs.198

Question: 14

Twenty cylindrical pillars of the Parliament House are to be cleaned. If the diameter of each pillar is 0.50m and height is 4m.What will be the cost of cleaning them at the rate of Rs 2.50 per square meter?

Solution:

Diameter of each pillar = 0.5,

Radius of each pillar(r) = d/2

= 0.5/2 = 0.25 m

Height of each pillar = 4 m

Lateral surface area of one pillar

= 2πrh = 2 * 3.14 * 0.25 * 4 = 44/7m²

Lateral surface area of 20 pillars = 20 * 44/7 m²

Cost of cleaning one pillar = Rs 2.50 per square meter

Cost of cleaning 20 pillars = Rs 2.50 * 20 * 44/7 m² = Rs. 314.28