find the altitude of a right circular cone of maximum curved surface area which can be inscribed in a sphere of radius R
Answers
Given : a right circular cone of maximum curved surface area inscribed in a sphere of radius R
To Find : altitude of a right circular cone
Solution:
Let say altitude = h
h = R + x => x = h - R
R = Radius of Sphere
Radius of cone r =√( R² - x²) = =√( R² - (h - R)²) = √(2R - h)h
=> r² = (2R - h) h = 2Rh - h²
L² = h² + r²
= L² = h² + 2Rh - h² = 2Rh
curved surface area A = πrL
Z = A² as A is + ve Hence A will be maximum when Z is maximum
Z = π²r²L²
=> Z =π²(2Rh - h²)2Rh
=> Z =2Rπ²(2Rh² - h³)
dZ/dh = 2Rπ² ( 4Rh - 3h²)
dZ/dh = 0
=> 4Rh - 3h² = 0
=> h = 4R/3
dZ/dh = 2Rπ² ( 4Rh - 3h²)
d²Z/dh² = 2Rπ² ( 4R - 6h)
h = 4R/3
=> d²Z/dh² = 2Rπ² ( 4R - 8R )
= - 8R²π² < 0
Hence Z is maximum at h = 4R/3
altitude of a right circular cone of maximum curved surface area which can be inscribed in a sphere of radius R is 4R/3
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