A given right cone has volume `p ,` and the largest right circular cylinder that can be inscribed in the cone has volume `qdot` Then `p : q` is (a) 9:4 (b) 8:3 (c) 7:2 (d) none of these

Options

A. `9:4`
B. `8:3`
C. `7:2`
D. none of these

Answer

1

Explanation

Let H be the height of the cone and `alpha` be its semi verticle angel suppose that x is the radius of the inscribed cylinder and h is its height
`h=QL=OL-OQ=H-x cot alpha`
V=Volume of the cylinder =`pi x^(2)(H-x cot alpha)`
Also p=`1/3 pi(H tan alpha)^(2)H`
`(dV)/(dx)=pi(2Hx-3x^(2)cot alpha)`
So ,`(dv)/(dx)=0i.e x =0 or x =2/3 H tan alpha`
`(d^(2)v)/(dx^(2))|_(x=2/3H tan alpha=-2pihlt0`
Sop V is maximum when x =`2/3 H tan alpha`
`q=v_(max)=pi(4)/(9)H^(2)tan^(2)alpha(1)/(3)H`
`=(4)/(27)(3pip tan^(2)alpha)/(pitan^(2)alpha)=4/9p`
Hene p:q=9:4