## 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 x r = 132

⇒ r = cm = 21cm

Volume of a vessel (h) = πr^{2}h

= x (21)^{2} x 25cm^{3}

= x 21 x 21 x 25cm^{3}

= 22 x 3 x 21 x 25 cm^{3}

= 34650 cm^{3}

⇒ 34650 cm^{3} = 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) = cm = 12cm

Outer radius of the pipe(R) = 14cm

Length of the pipe (h) = 35 cm

Volume of cylindrical pipe = π (R^{2} – r^{2})h.

= (14^{2 – }12^{2}) x 35 = 22 x (196 – 144) x 5

=22 x 52 x 5 = 5720 cm^{3}

Mass of wood in the pipe = 5720 x 0.6 = 3432 g = 3.432 kg (1 cm^{3} 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 cm^{3} = 300 cm^{3}

(ii) For cylindrical pack,

∴ Radius (r) = cm

Height (h) = 10 cm

∴ Volume = πr^{2}h = x ()^{2} x 10cm

= x x x 10cm

= 11 x 7 x 5cm^{3} = 385cm^{3}

∴ Capacity of the cylindrical pack = 385 cm^{3}

So, the cylindrical pack has greater capacity

by (385 – 300) cm^{3} = 85 cm^{3}

**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 = πr^{2}h

⇒ Volume of given cylinder

=3.14 (3)^{2} x 5cm^{3}

= 3.14 x 3 x 3 x 5 cm^{3}

= cm = 141.3cm^{3}

Thus, the required volume = 141.3 cm^{3}

**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 m^{2} = ₹ 20

Total cost

∴ Inner curved surface area of the vessel = 110 m^{2}

(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 cm^{3} [1 litre = 10(x) cm^{3}]

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 m^{2}.

**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 (R^{2} – r^{2})h.

= [(0.35)^{2} – ^{ }(0.05)^{2}] x 14 = x(0.1225 – 0.0025) x 2 = 22 x 0.12 x 2 = 5.28 cm^{3}

∴ 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 = πR^{2}h

= (0.05)^{2} x 14

= 22 x 0.0025 x 2

= 0.11 cm^{3}

**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 = πr^{2}h.

= x (3.5)^{2} x 4

= x 2.5 x 3.5 x 4 = 22 x 0.5 x 3.5 x 4 = 154 cm^{3}

∴The volume of soup per day for 250 patient = 250 x154 = 38500 cm^{3} or 38.5 liters