SURFACE AREA AND VOLUME
EXERCISE 13.6
EXERCISE 13.6
Solution: Let the base radius of the cylindrical vessel be ‘r’ cm.
∴ Circumference = 2πr
⇒ 2πr = 132 [∵Circumference = 132 cm]
∵ Capacity of the vessel = Volume of the vessel
∴ Capacity of cylindrical vessel = 34650 cm3
Since 1000 cm3 = 1 litre
Thus, the vessel can hold 34.65 l of water.
Solution: Inner diameter of the cylindrical pipe = 24 cm
Length of the pipe (h) = 35 cm
∵ Inner volume of the pipe = πr2h
Outer volume of the pipe = πR2h
∴ Amount of wood (volume) in the pipe = Outer volume – Inner volume
= πR2h – πr2h
= πh(R2 – r)
= πh(R + r)(R – r) [∵ a2 – b2 = (a + b)(a – b)]
Mass of the wood in the pipe = [Mass of wood in 1 m3 of wood] × [Volume of wood in the pipe]
= [0.6g] × [22 × 5 × 26 × 2] cm3
= 6 × 22 × 10 × 26g
= 6 × 22 × 26 g = 3432 g
Thus, the required mass of the pipe is 3.432 kg.
Q 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?
Solution:
Length (l) = 5 cm
Breadth (b) = 4 cm
Height (h) = 15 cm
∴ Volume = l × b × h = 5 × 4 × 15 cm3 = 300 cm3
⇒ Capacity of the rectangular pack = 300 cm3 …(1)
For cylindrical pack:
Base diameter = 7 cm
Q 4. If the lateral surface of a cylinder is 94.2 cm2 and its height is 5 cm, then find: (1) radius of its base (ii) its volume. (Use π = 3.14)
Solution: Height of the cylinder (h) = 5 cm
Let the base radius of the cylinder be ‘r’.
(i) Since lateral surface of the cylinder = 2 πrh
But lateral surface of the cylinder = 94.2 cm2
⇒ 2πrh = 94.2
Thus, the radius of the cylinder = 3 cm
(ii) Volume of a cylinder = πr2h
Volume of the given cylinder = 3.14 × (3)2 × 5 cm3
Thus, the required volume = 141.3 cm3
Q 5. It costs Rs. 2200 to paint the inner curved surface of cylindrical vessel 10 m the cost of painting is at the rate of Rs. 20 per m2; find:
(i) inner curved surface of the vessel
(ii) radius of the base
(iii) capacity of the vessel.
Solution: (i) To find inner curved surface
Total cost of painting = Rs. 2200
Rate of painting = Rs. 20 per m2
Inner curved surface of the vessel = 110 m2
(ii) To find radius of the base
Let the base radius of the cylindrical vessel.
∵ Curved surface of a cylinder = 2 πrh
∴ 2πrh = 110
The required radius of the base = 1.75 m
(iii) To find the capacity of the vessel
Since, volume of a cylinder = πr2h
Q 6. The capacity of 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 l
= 15.4 × 1000 cm3
Q 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 graphite is I mm. If the length of the pencil is 14 cm, find the volume of the wood and that of the graphite.
Thus, the required volume of the graphite = 0.11 cm3
For the pencil
Volume of the wood = [Volume of the pencil] – [Volume of the graphite]
= 5.39 cm3 – 0.11 cm3
= 5.28 cm3
Thus, the required volume of the wood is 5.28 cm3.
Q 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: The bowl is cylindrical.
Diameter of the base = 7 cm
Thus, the hospital needs to prepare 38.5 litres of soup daily for 250 patients.