Class 9 - Mathematics
Surface Areas and Volumes - Exercise 13.6
Top Block 1
Exercise 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? (1000 cm3 = 1l)
Answer :
Here, h = 25 cm, 2πr = 132 cm.
Given, the circumference of the base of a cylindrical vessel is 132 cm
⇒ 2πr = 132
⇒ 2 * (22/7) * r = 132
⇒ r = (132 * 7)/(2 * 22)
⇒ r = (132 * 7)/44
⇒ r = 3 * 7
⇒ r = 21 cm
Now, volume of the cylinder = πr2 h
= (22/7) * (21)2 * 25
= (22/7) * 21 * 21 * 25
= 22/ * 3 * 21 * 25
= 34650 cm3
= 34650/1000 litres [Since 1000 cm3 = 1l]
= 34.65 litres
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.
Answer :
Here, inner radius (r) = 24/2 = 12 cm
Outer radius (R) = 28/2 = 14 cm, h = 35 cm
Volume of the wood used in the pipe = π(R2 – r2) h
= (22/7) * (142 – 122) * 35
= 22 * (196 – 144) * 5
= 22 * 52 * 5
= 5720 cm3
Mass of 1 cm3 of wood = 0.6 g
So, Mass of 5720 cm3 of wood = 0.6 * 5720 g
= 3432 g
= 3432/1000 kg [Since 1000 g = 1 kg]
= 3.432 kg
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 and
(ii) a plastic cylinder with circular base of diameter 7 cm and height 10 cm. Which container has greater capacity and by how much?
Answer :
For tin can with rectangular 6 base.
l = 5 cm, b = 4 cm, h = 15 cm
Volume of the tin can = lbh = 5 * 4 * 15 = 300 cm3
For plastic cylinder with circular base.
r = 7/2 cm = 3.5 cm, h = 10 cm
Volume of the plastic cylinder = πr2 h
= (22/7) * 3.5 * 3.5 × 10
= 22 * 3.5 * 0.5 * 10
= 385 cm3
Difference in the capacities of the two containers = 385 – 300 = 85 cm3
Hence, the plastic cylinder with circular base has greater capacity by 85 cm3
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)
Answer :
Here, h = 5 cm, 2πrh = 94.2 cm2
(i) 2πrh = 94.2
⇒ 2 * 3.14 * r * 5 = 94.2
⇒ r = 94.2/(2 * 3.14 * 5)
⇒ r = 94.2/(10 * 3.14)
⇒ r = 94.2/31.4
⇒ r = 3
Hence, base radius of the cylinder = 3 cm
(ii) Volume of the cylinder = πr2 h
= 3.14 * 3 * 3 * 5
= 141.3 cm3
Question : 5: It costs Rs 2200 to paint the inner curved surface of a cylindrical vessel 10 m deep. If the cost of painting is at the rate of Rs 20 per m2, find
(i) Inner curved surface area of the vessel (ii) radius of the base (iii) capacity of the vessel.
Answer :
Here, h = 10 m
(i) Inner curved surface area = Total cost of painting per m2
= 2200/20
= 110 m2
(ii) We have, 2πrh = 110
⇒ 2 * (22/7) * r * 10 = 110
⇒ r = (110 * 7)/(2 * 22 * 10)
⇒ r = 770/440
⇒ r = 1.75 m
(iii) Capacity of the vessel = πr2 h
= (22/7) * 1.75 * 1.75 * 10
= 22 * 1.75 * 0.25 * 10
= 96.25 m3
= 96.25 kl [Since 1 m3 = 1 kl]
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?
Answer :
Here, h = 1 m,
Given, volume = 15.4 litres
= 15.4/1000 m3
= 0.0154 m3
Also, volume of the cylindrical vessel = πr2 h
⇒ 0.0154 = (22/7) * r2 * 1
⇒ r2 = (0.0154 * 7)/22
⇒ r2 = 0.0049
⇒ r = √(0.0049)
⇒ r = 0.07 m
So, total surface area of the cylinder = 2πr (h + r)
= 2 * (22/7) * 0.07 (1 + 0.07)
= 44 * 0.01 * 1.07
= 0.4708 m2
Hence, 0.4708 m2 of metal sheet would be needed.
Mddle block 1
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.
Answer :
Here, h = 14 cm.
Radius of the pencil (R) = 7/2 mm = 0.35 mm.
Radius of the graphite (r) = 1/2 mm = 0.05 mm.
Volume of the graphite = πr2 h
= (22/7) * 0.05 * 0.05 * 14
= 22 * 0.05 * 0.05 * 2
= 0.11 mm3
Volume of the wood = π(R2 – r2)h
= (22/7) * [(0.35)2 – (0.05)2] * 14
= 22 * [0.1225 – 0.0025] * 2
= 22 * 0.12 * 2
= 5.28 cm3
Hence, volume of the wood = 5.28 cm3 and volume of the graphite = 0.11 cm3
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?
Answer :
Here, r = 7/2 cm = 3.5 cm, h = 4 cm
Capacity of 1 cylindrical bowl = πr2 h
= (22/7) * 3.5 * 3.5 * 4
= 22 * 3.5 * 0.5 * 4
= 154 cm3
Hence, soup consumed by 250 patients per day = 250 * 154 = 38500 cm3