Theory question 112
Mensuration of SolidsA water reservoir in the form of a cone mounted on a hemisphere is built such that the plane face of the hemisphere fits exactly to the base of the cone and the height of the cone is 6 times the radius of its base.
(a) Illustrate this information in a diagram.
(b) If the volume of the reservoir is 333⅓ π m3 calculate, correct to the nearest whole number, the: (i) volume of the hemisphere; (ii) total surface area of the reservoir. [Take π = 22/7]
Show the model answer
(a) A cone of base radius r and height 6r standing on the flat face of a hemisphere of radius r.
(b) Volume = ⅔πr³ + ⅓πr²(6r) = (8/3)πr³ = (1000/3)π → r³ = 125 → r = 5 m
(i) Hemisphere = ⅔ × 22/7 × 125 ≈ 261.9 ≈ 262 m³
(ii) Slant height l = √(5² + 30²) = √925 ≈ 30.41 m
Curved surface of hemisphere = 2πr² = 2 × 22/7 × 25 ≈ 157.1 m²
Curved surface of cone = πrl = 22/7 × 5 × 30.41 ≈ 477.9 m²
Total surface area ≈ 635 m²
Find r first from the total volume.