A solid hemisphere is attached to the top of a cylinder, having the same radius as that of the cylinder. If the height of the cylinder were doubled (keeping both radii fixed), the volume of the entire system would have increased by 50%. By what percentage would the volue have increased if the radii of the hemisphere and the cylinder were doubled (keeping the height fixed)?

Options

A. 3
B. 4
C. 5
D. 6

Answer

C

Explanation

Let hight of radius of cylinder are h & r respectively.
Then volume `V_(1)=pir^(2)h+2/3pir^(3)" …(1)"`
When height of cylinder is doubled then volume `V_(2)=pir^(2)h+2/3pir^(3)" …(2)"`
Given that `(V_(2))/(V_(1))=3/2rArr(2h+(2)/(3)r)/(h+2/3r)=3/2`
`rArr2h+2/3r=3/2h+r`
`rArrh/2=r/3rArrh=2/3r" ...(3)"`
When radius is doubled then volume `V_(2)^(1)=4pir^(2)h+16/3pir^(3)`
`(V_(2)^(1))/(V_(1))=(4h+16/3r)/(h+2/3r)`
By `(3)(V_(2)^(1))/(V_(1))=(4h+8h)/(h+h)=6`
Hence volume will be increased by 500%