a hemisphere of radius 8cm is cast into a right circular come of Base radius 6cm. determine the height of the cone, correct to two places of decimal
Answers
Answer:
h = 25.3 cm - closed
h = 20.47 cm - open
h = 28.44 cm - filled
Step-by-step explanation:
A hemisphere of lead of radius 8cm is cast into a right circular cone of base
radius 6cm. Determine the height of the cone.
Hemisphere & cone can be closed or opened
Case 1 : when both are closed
Case 2 : when both are opened
Case 3 : when both are filled
Case 1 : when both are closed
Surface Area of closed Hemisphere = 2πR² + πR² = 3πR²
Surface Area = 3 * π * (8)² = 192π cm²
Surface Area of closed Cone = πR² + πR(√(h² + r²))
= π (6)² + π6 (√(h² + 6²))
= π6 ( 6 + √(h² + 6²) )
Surface area would be same as it is made from that material only
192π = π6 ( 6 + √(h² + 6²) )
=> 32 = 6 + √(h² + 36) )
=> 26 = √(h² + 36)
squaring both sides
=> 676 = h² + 36
=> h² = 640
=> h = 25.3 cm
Case 2 : when both are opened
Surface Area of open Hemisphere = 2πR²
Surface Area = 2 * π * (8)² = 128π cm²
Surface Area of open Cone = πR(√(h² + r²))
= π6 (√(h² + 6²))
Surface area would be same as it is made from that material only
128π = π6√(h² + 6²) )
=> 64/3 = √(h² + 36) )
squaring both sides
=> 4096/9 = h² + 36
=> h² = (4096 - 324)/9
=> h² = 3772/9
=> h = 61.42/3
=> h = 20.47 cm
Case 3 : when both are filled
if both are filled then volume of both will be equal
Volume of hemisphere = (2/3)πR³ = (2/3)π(8)³ = 1024π/3
Volume of Cone = (1/3)πR²h = (1/3)π(6)²h = 36πh/3
36πh/3 = 1024π/3
h = 1024/36
=> h = 28.44 cm
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