A hemispherical solid of radius 21 cm is melted and made into a cylinder whose height is 14 cm. The volume of the cylinder is given by (a × ba × cb)π, where a, b, and c are prime. Then the value of (a+b)c is:
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Correct answer: A
Explanation
The correct answer is option 1, i.e., 35.
Volume of hemisphere = (2 × π × r3) ÷ 3
r = 21 cm
Volume = 2 × π × 213 ÷ 3 = 2 × π × 9261 ÷ 3 = 18522 ÷ 3 × π = 6174 π cm3
Volume of cylinder = π × r2 × h = 6174 π
h = 14 → π × r2 × 14 = 6174 π
r2 × 14 = 6174 → r2 = 6174 ÷ 14 = 441 → r = 21 cm
Cylinder volume = 6174 π = 2 × 3 × 72 × 33 × π? Let's factor:
6174 = 2 × 3 × 1029 → 1029 = 3 × 343 → 343 = 73
So 6174 = 2 × 32 × 73
Volume = 2 × 32 × 73 × π → a = 2, b = 3, c = 7
(a + b) × c = (2 + 3) × 7 = 5 × 7 = 35