Chapter 11: Surface Areas and Volumes

Circumference of the base of cylindrical vessel = 132 cm

Height of vessel, h = 25 cm

Let r be the radius of the cylindrical vessel.

Step 1: Find the radius of the vessel.

We know that the circumference of the base = 2πr, so

2πr = 132 (given)

r = (132/(2 π))

r = 66×7/22 = 21

The radius is 21 cm.

Step 2: Find the volume of the vessel.

Formula: Volume of cylindrical vessel = πr²h

= (22/7)×21²×25

= 34650

Therefore, the volume is 34650 cm³

Since 1000 cm³ = 1L

So, Volume = 34650/1000 L= 34.65L

Therefore, the vessel can hold 34.65 litres of water.

Inner radius of cylindrical pipe, say r₁ = diameter₁/ 2 = 24/2 cm = 12cm

Outer radius of cylindrical pipe, say r₂ = diameter₂/ 2 = 28/2 cm = 14 cm

Height of pipe, h = Length of pipe = 35cm

Now, the Volume of pipe = π(r₂²-r₁²)h cm³

Substitute the values.

Volume of pipe = 110×52 cm³ = 5720 cm³

Since Mass of 1 cm³ wood = 0.6 g

Mass of 5720 cm³ wood = (5720×0.6) g = 3432 g or 3.432 kg.

(i) Tin can will be cuboidal in shape.

Dimensions of the tin can are

Length, l = 5 cm

Breadth, b = 4 cm

Height, h = 15 cm

Capacity of tin can = l×b×h= (5×4×15) cm³ = 300 cm³

(ii) Plastic cylinder will be cylindrical in shape.

Dimensions of the plastic can are

Radius of the circular end of plastic cylinder, r = 3.5cm

Height , H = 10 cm

Capacity of the plastic cylinder = πr²H

Capacity of the plastic cylinder = (22/7)×(3.5)²×10 = 385

Capacity of the plastic cylinder is 385 cm³

From the results of (i) and (ii), the plastic cylinder has more capacity.

Difference in capacity = (385-300) cm³ = 85cm³

CSA of cylinder = 94.2 cm²

Height of cylinder, h = 5cm

(i) Let the radius of the cylinder be r.

Using the CSA of the cylinder, we get

2πrh = 94.2

2×3.14×r×5 = 94.2

r = 3

The radius is 3 cm.

(ii) Volume of cylinder

The formula for the volume of the cylinder = πr²h

Now, πr²h = (3.14×(3)²×5) (using the value of r from (i))

= 141.3

Volume is 141.3 cm³

(i) Rs 20 is the cost of painting 1 m² area.

Rs 1 is the cost to paint 1/20 m² area.

So, Rs 2200 is the cost of painting = (1/20×2200) m²

= 110 m² area

The inner surface area of the vessel is 110m².

(ii) Radius of the base of the vessel, let us say r.

Height (h) = 10 m and

Surface area formula = 2πrh

Using the result of (i),

2πrh = 110 m²

2×22/7×r×10 = 110

r = 1.75

The radius is 1.75 m.

(iii) Volume of vessel formula = πr²h

Here r = 1.75 and h = 10

Volume = (22/7)×(1.75)²×10 = 96.25

The volume of vessel is 96.25 m³

Therefore, the capacity of the vessel is 96.25 m³ or 96250 litres.

Height of cylindrical vessel, h = 1 m

Capacity of cylindrical vessel = 15.4 litres = 0.0154 m³

Let r be the radius of the circular end.

Now,

Capacity of cylindrical vessel = (22/7)×r²×1 = 0.0154

After simplifying, we get r = 0.07 m

Again, the total surface area of the vessel = 2πr(r+h)

= 2×22/7×0.07(0.07+1)

= 0.44×1.07

= 0.4708

Total surface area of the vessel is 0.4708 m²

Therefore, 0.4708 m² of the metal sheet would be required to make the cylindrical vessel.

Radius of pencil, r₁ = 7/2 mm = 0.7/2 cm = 0.35 cm

Radius of graphite, r₂ = 1/2 mm = 0.1/2 cm = 0.05 cm

Height of pencil, h = 14 cm

Formula to find the volume of wood in pencil = (r₁²-r₂²)h cubic units

Substituting values, we have,

= [(22/7)×(0.35²-0.05²)×14]

= 44×0.12

= 5.28

This implies that the volume of wood in pencil = 5.28 cm³

Again,

Volume of graphite = r₂²h cubic units

Substituting the values, we have,

= (22/7)×0.05²×14

= 44×0.0025

= 0.11

So, the volume of graphite is 0.11 cm³.

Diameter of the cylindrical bowl = 7 cm

Radius of the cylindrical bowl, r = 7/2 cm = 3.5 cm

Bowl is filled with soup to a height of4cm, so h = 4 cm

Volume of the soup in one bowl= πr²h

(22/7)×3.5²×4 = 154

Volume of the soup in one bowl is 154 cm³

Therefore,

Volume of the soup given to 250 patients = (250×154) cm³= 38500 cm³

= 38.5litres.