Question

    The curved surface area of a cone exceeds that of a

    cylinder by __ cm². The radius of the cone is twice that of the cylinder. The area of the cylinder's base is __ cm². The respective heights of the cone and the cylinder are _____ and _____.   (take π = 3) The values given in which of the following options will fill the blanks in the same order to make the above statement true (I) 12 cm2, 12, 3cm and 6 cm (II) 24 cm2, 27, 8 cm and 7 cm (III) 120 cm2, 48, 15 cm and 12 cm.
    A Only I Correct Answer Incorrect Answer
    B Only II Correct Answer Incorrect Answer
    C Only I and II Correct Answer Incorrect Answer
    D Only I and III Correct Answer Incorrect Answer
    E All I, II and III Correct Answer Incorrect Answer

    Solution

    Let the radius of the cone and cylinder be 'r' cm and 'R' cm and the lateral height of the cone be 'L' cm From (I) , The base area of the cylinder = 12 cm2 So, π X R2 = 12 So, R2 = (12/3) = 4 cm So, 'R' = 2 cm So, the radius of the cone = 2 X 2 = 4 cm Given the height of the cone and cylinder is 3 cm and 6 cm So, the lateral height of the cone (L) = √(3) 2 + (4) 2 = √25 = 5 cm So, the curved surface area of the cone = π X radius X lateral length = 3 X 4 X 5 = 60 cm2 The curved surface area of the cylinder = 2 X π X radius X height = 2 X 3 X 2 X 6 = 72 cm2 The difference between the curved surface area of cone and cylinder = 72 - 60 = 12 cm2 So, (I) is true. From (II) , The base area of the cylinder = 27 cm So, π X R2 = 27 So, R2 = (27/3) = 9 cm So, 'R' = 3 cm So, the radius of the cone = 2 X 3 = 6 cm Given the height of the cone and cylinder is 8 cm and 7 cm So, the lateral height of the cone (L) = √(6) 2 + (8) 2 = √100 = 10 cm So, the curved surface area of the cone = π X radius X lateral height = 3 X 6 X 10 = 180 cm2 The curved surface area of the cylinder = 2 X π X radius X height = 2 X 3 X 3 X 7 = 126 cm2 The difference between the curved surface area of cone and cylinder = 180 - 126 = 54 cm2 So, (II) is not true. From (III) , The base area of the cylinder = 48 cm So, π X R2 = 48 So, R2 = (48/3) = 16 cm So, 'R' = 4 cm So, the radius of the cone = 2 X 4 = 8 cm Given the height of the cone and cylinder is 15 cm and 12 cm So, the lateral height of the cone (L) = √(15) 2 + (8) 2 = √289 = 17 cm So, the curved surface area of the cone = π X radius X lateral height = 3 X 8 X 17 = 408 cm2 The curved surface area of the cylinder = 2 X π X radius X height = 2 X 3 X 4 X 12 = 288 cm2 The difference between the curved surface area of cone and cylinder = 408 - 288 = 120 cm2 So, (III) is true.

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