A bell tent consists of a conical portion above a cylindrical portion near the ground. For a given volume and a circular base of a given radius, the amount of the canvas used is a minimum when the semi-vertical angle of the cone is `cos^(-1)2/3` (b) `sin^(-1)2/3` `cos^(-1)1/3` (d) none of these

Options

A. `cos^(-1)2//3`
B. `sin^(-1)2//3`
C. `cos^(-1)1//3`
D. none of these

Answer

1

Explanation

Given volume and r
Now V= volume of cone + volume of cylinder
`=(pi)/(3)r^(2)h+pir^(2)H`
`=(pi)/(3)r^(2)(h+3H)`
`H=(3V)/(pir^(2))-h(3)`
Now surface area S= `pirl+2pirH`
`=pi r sqrt(h^(2)+r^(2)+2pirxx((3V)/(pir^(2))-h)/(3))`
`let (ds)/(dh)=0 pir(h)/sqrt(h^(2)+r^(2))-(2pir)/(3)=0`
`(h)/sqrt(h^(2)+r^(2)) =2/3 or 5h^(2)=4r^(2)or (r)/(h) =sqrt(5)/(2)=tan theta`
`cos theta =2/3 or theta = cos^(-1)2/3`