Q 1. A conical vessel of a given storage capacity V is to be constructed from a thin metal sheet. Show that the vessel will require least material when its height is √2 times the radius of its base.
Answers
Answered by
1
height is √2 times the radius of its base for least material when top id open of conical vessel
Step-by-step explanation:
Volume of a Cone = (1/3)πR²h
V = (1/3)πr²h
Let say Radius = r
Then h = 3V/πr²
Surface Area = πr√r² + h² ( as Vessel must be opened from top)
S = πr√r² + (3V/πr²)²
S = πr√(r² + 9V²/π²r⁴)
S = πr√(π²r⁶ + 9V²)/πr²
S = √(π²r⁶ + 9V²) / r
find ds/dr and put dS/dr = 0
=> 2π²r⁶ = 9V²
2π²r⁶ = 9( (1/3)πr²h)²
=> 2r² = h²
=> h = √2r
height is √2 times the radius of its base
Similar questions