Class IX Math NCERT Solution for Surface Areas and Volumes 13.6

XERCISE 13.6 (Page 230)
Q1.   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 = 1 l)
Sol: 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.
Q2.   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 l cm3 of wood has a mass of 0.6g.
Sol: Here,
           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.
Q3.   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?
Sol: For rectangular pack:
                 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
           
Q4.   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)
Sol: 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
Q5.   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.
Sol: (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
                 
                 
Q6.   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?
Sol: Capacity of the cylindrical vessel = 15.4 l
                 = 15.4 × 1000 cm3
           
Q7.   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 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.
Q8.   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?
Sol: 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.

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