A given right circular cone has a volume p, and the largest right circular cylinder that can be inscribed in the cone has a volume q. Then p : q is -
Text Solution
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Let H be the height of the cone and α be its semi vertical angle. Suppose that x is the radius of the inscribed cylinder and h be its height h = QL = OL – OQ = H – x cot α
V = volume of the cylinder = π x 2 (H – x cot α )
Also p = 
π (H tan α ) 2 H (i)
= π (2Hx – 3x 2 cot α )
So 
= 0 ⇔ x = 0, x = 
H tan α
= – 2 π h < 0.
So V is maximum when x = 
H tan α and
q = V max = 
H 2 tan 2 α 
H
= 
p. [using (i)]
Hence p : q = 9 : 4.
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