The curved surface area of a cone exceeds that of a

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.