FINDING VOLUME AND SURFACE AREA OF CYLINDERS

A cylinder is a three dimensional solid that holds two parallel bases joined by a curved surface at a fixed distance.

Lateral surface area of cylinder = 2πrh

Total surface area = 2πr(h + r)

Volume of cylinder = πr²h

What is surface area of cylinder ?

The surface area of a cylinder can be defined as the total space covered by the flat surfaces of the bases of the cylinder and its curved surface.

Difference between lateral and total surface area :

Lateral surface area is the area around the shape excluding top and bottom.

Total surface area is the area including top and bottom.

What is volume of cylinder ?

The volume of a cylinder is the density of the cylinder which signifies the amount of material it can carry or how much amount of any material can be immersed in it.

Problem 1 :

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

Solution :

Diameter = 7 cm, Radius = 3.5 cm and Height = 12 cm

Volume of milk in 1 glass = πr²h

= π × (3.5)² × 12

= 22/7 × 12.25 × 12

= 462 cm³

For 1600 students milk needed is

= 1600 × 462

= 739200

1 cm³ = 1000 litres

= 739200/1000

= 739.2 litres

Therefore, the quantity of milk required is 739.2 litres.

Problem 2 :

If the radius of a right circular cylinder is decreased by 50% and its height is increased by 60%, its volume will be decreased by

a) 10%    b) 60%    c) 40%    d) 20%

Solution :

Let the original measures be 100%.

Radius of new cylinder = 50% of r ==> (50r/100)

height of the new cylinder = 160% of h ==> (160h/100)

Volume = π (50r/100)² × (160h/100)

= 40% of πr²h

So, 60% of decreased.

So, option (b) is correct.

Problem 3 :

The ratio of radii of two cylinders is 1:2 and the heights are in the ratio 2:3. Find the ratio of their volumes.

Solution :

Let, V₁ be the volume of first cylinder

V₂ be the volume of first cylinder

V₁ = πr₁²h₁

V₂ = πr₂²h₂

V₁/V₂ = (r₁/r₂)² × h₁/h₂

= (1/2)²× 2/3

= 1/4 × 2/3

= 1/6

So, the ratio is 1 : 6.

Problem 4 :

If the capacity of a cylindrical tank is 1848 m³ and the diameter of its base is 14 m. find the depth of the tank.

Solution :

Volume of the cylinder = 1848 m³

Diameter of the base = 14 m

Radius r = d/2

= 14/2

= 7 cm

Volume of the cylinder = πr²h

1848 = (22/7) ⋅ 7 ⋅ 7 ⋅ h

1848 = 154 h

h = 1848/154

h = 12 m

Therefore, the depth of the tank is found to be 12 m.

Problem 5 :

The radius of the base of a cylinder is 39 in. and its height is 33 in. Find the surface area of the cylinder in terms of π.

Solution :

Radius = 39 inches and height = 33 inches

Surface area of cylinder = 2 π r h

= 2 x π x 39 x 33

= 2574π square inches

Problem 6 :

Andrew is planning to cover the lateral surface of a large cylindrical garbage can with decorative fabric for a theme party. The can has a diameter of 3 feet and a height of 3.5 feet. How much fabric does he need if he covers the lid but not the bottom of the can?

Solution :

Radius = 3/2 ==> 1.5 feet

height = 3.5 feet

Surface area of cylinder = 2 π r h

= 2 x π x 1.5 x 3.5

= 10.5 π square feet

Problem 7 :

Find the surface area of the composite figure.

Solution :

Surface area of rectangular prism + surface area of cylinder/2

= 2(11 x 6) + (11 x 9) + (2 π x 4.5 x 11)/2

= 132 + 99 + 155.43

= 386.43

So, the surface area of the composite figure is 386.43 square meter

Problem 8 :

How much salsa is missing from the jar?

Solution :

Quantity of salsa = πr² h

r = 5 cm and height = 4 cm

= 3.14 x 5² x 4

= 314 cm²

Problem 9 :

About how many gallons of water does the water cooler bottle contain? (1 ft³ ≈ 7.5 gal)

a) 5.3 gal     b) 10 gal     c)  17 gal      d) 40 gal

Solution :

Quantity of water = πr² h

r = 0.5 ft and height = 1.7 ft

= 3.14 x 0.5² x 1.7

= 1.3345 cubic ft

1 ft³ ≈ 7.5 gal

Converting into gallons,

= 1.3345 x 7.5

= 10.008 gallons

Approximately 10 gallons, option b is correct.

Problem 10 :

A cylindrical swimming pool has a diameter of 16 feet and a height of 4 feet. About how many gallons of water can the pool contain? Round your answer to the nearest whole number. (1 ft³ ≈ 7.5 gal)

Solution :

Quantity of water = πr² h

Radius = 8 ft and height = 4 ft

= 3.14 x 8² x 4

= 803.84 cubic ft

1 ft³ ≈ 7.5 gal

Converting into gallons,

= 803.84 x 7.5

= 6028.8 gallons

Approximately 6029 gallons.