NCERT solutions for class 9 Maths chapter 13 Surface Areas and Volumes (Exercise

NCERT solutions for class 9 Maths

Surface Areas and Volumes

Chapter 13 Exercise 13.6

EX 13.6 QUESTION 1.

The circumference of the base of a cylindrical vessel is 132 cm and its height is 25 cm. How many litres of water can it hold? (1000cm³ =1 L.)
Solution:

Circumference of the base of cylindrical vessel C = 132 cm , height h = 25 cm

Let the radius of the cylindrical vessel = r cm

Circumference of the base = 2πr
⇒ 2πr = 132 (Circumference = 132 cm)
= 2 x $\frac { 22 }{ 7 }$ x r = 132
⇒ r = $\frac { 132 x 7 }{ 2 x 22 }$ cm = 21cm
Volume of a vessel (h) = πr²h
= $\frac { 22 }{ 7 }$ x (21)² x 25cm³
= $\frac { 22 }{ 7 }$ x 21 x 21 x 25cm³
= 22 x 3 x 21 x 25 cm³
= 34650 cm³
⇒ 34650 cm³ = 34.65 litres

Ex 13.6 Question 2.

The inner diameter of a cylindrical wooden pipe is 24 cm and its outer diameter is 28 cm. The length of the pipe is 35 cm. Find the mass of the pipe, if 1 cm3 of wood has a mass of 0.6 g.
Solution:
Inner radius of cylindrical pipe (r) = $\frac { 24 }{ 2 }$ cm = 12cm
Outer radius of the pipe(R) = 14cm
Length of the pipe (h) = 35 cm
Volume of cylindrical pipe = π (R² – r²)h.
= $\frac { 22 }{ 7 }$ (14^{2 –}12²) x 35 = 22 x (196 – 144) x 5
=22 x 52 x 5 = 5720 cm³
Mass of wood in the pipe = 5720 x 0.6 = 3432 g = 3.432 kg (1 cm³ of wood has mass of 0.6g.)

Ex 13.6 Question 3.

A soft drink is available in two packs
(i) a tin can with a rectangular base of length 5 cm and width 4 cm, having a height of 15 cm.
(ii) a plastic cylinder with circular base of diameter 7 cm and height 10 cm. Which container has greater capacity and by how much?
Solution:
(i) For rectangular pack,
Length (l) = 5 cm,
Breadth (b) = 4 cm
Height (h) = 15 cm
Volume = l x b x h = 5 x 4 x 15 cm³ = 300 cm³

(ii) For cylindrical pack,
∴ Radius (r) = $\frac { 7 }{ 2 }$ cm
Height (h) = 10 cm
∴ Volume = πr²h = $\frac { 22 }{ 7 }$ x ( $\frac { 7 }{ 2 }$ )² x 10cm
= $\frac { 22 }{ 7 }$ x $\frac { 7 }{ 2 }$ x $\frac { 7 }{ 2 }$ x 10cm
= 11 x 7 x 5cm³ = 385cm³
∴ Capacity of the cylindrical pack = 385 cm³
So, the cylindrical pack has greater capacity
by (385 – 300) cm³ = 85 cm³

Ex 13.6 Question 4.

If the lateral surface of a cylinder is 94.2 cm² and its height is 5 cm, then find
(i) radius of its base,
(ii) its volume. (Use π = 3.14)
Solution:
Height (h) = 5 cm
Let the base radius of the cylinder =r.

(i) Lateral surface area of the cylinder = 2πrh = 94.2

Thus, the radius of the cylinder = 3 cm

(ii) Volume of a cylinder = πr²h
⇒ Volume of given cylinder
=3.14 (3)² x 5cm³
= 3.14 x 3 x 3 x 5 cm³
= $\frac { 1413 }{ 10 }$ cm = 141.3cm³
Thus, the required volume = 141.3 cm³

Ex 13.6 Question 5.

It costs ₹2200 to paint the inner curved surface of a cylindrical vessel 10 m deep. If the cost of painting is at the rate of ₹20 per m², find
(i) inner curved surface area of the vessel,
(ii) radius of the base,
(iii) capacity of the vessel.
Solution:
(i) Total cost of painting = ₹ 2200
Cost of painting of area 1 m² = ₹ 20
Total cost

∴ Inner curved surface area of the vessel = 110 m²

(ii) Let the radius of the cylindrical vessel = r

Curved surface area of a cylinder = 2πrh

Ex 13.6 Question 6.

The capacity of a closed cylindrical vessel of height 1 m is 15.4 litres. How many square metres of metal sheet would be needed to make it?
Solution:
Capacity of the cylindrical vessel
= 15.4 litres = 15.4 x 1000 cm³ [1 litre = 10(x) cm³]

Let the radius of the base of the vessel = r

Now, total surface area of the cylindrical vessel

Thus, the required metal sheet = 0.4708 m².

Ex 13.6 Question 7.

A lead pencil consists of a cylinder of wood with a solid cylinder of graphite filled in the interior. The diameter of the pencil is 7 mm and the diameter of the graphite is 1 mm. If the length of. the pencil is 14 cm, find the volume of the wood and that of the graphite.
Solution:
Inner radius of wood in pencil r = 1/2 = 0.5 mm = 0.05 cm
Outer radius R = 7/2 = 3.5 mm = 0.35 cm and length h = 14 cm
Volume of wood use d in pencil = π (R² – r²)h.
= $\frac { 22 }{ 7 }$ [(0.35)² – (0.05)²] x 14 = $\frac { 22 }{ 7 }$ x(0.1225 – 0.0025) x 2 = 22 x 0.12 x 2 = 5.28 cm³
∴ Radius of graphite inside the wood r = 1/2 = 0.5 mm = 0.05 cm
Height of the pencil (h) = 14 cm
Volume of the pencil = πR²h
= $\frac { 22 }{ 7 }$ (0.05)² x 14
= 22 x 0.0025 x 2
= 0.11 cm³

Ex 13.6 Question 8.

A patient in a hospital is given soup daily in a cylindrical bowl of diameter 7 cm. If the bowl is filled with soup to a height of 4 cm, how much soup the hospital has to prepare daily to serve 250 patients?
Solution:
Radius r = 7/2 = 3.5 and height of soup inside the cylindrical bowl = 4 cm Volume of cylindrical bowl = πr²h.

= $\frac { 22 }{ 7 }$ x (3.5)² x 4

= $\frac { 22 }{ 7 }$ x 2.5 x 3.5 x 4 = 22 x 0.5 x 3.5 x 4 = 154 cm³

∴The volume of soup per day for 250 patient = 250 x154 = 38500 cm³ or 38.5 liters

NCERT solutions for class 9 Maths chapter 13 Surface Areas and Volumes (Exercise – 13.6)