The diameter of a right circular cylinder is increased by 20%. Find the percentage decrease in its height if its volume remains unchanged.

Explanation

Let height of cylinder = 100 units
and diameter `= D rArr` radius `=(D)/(2)`
Volume of cylinder `V_(1)=pi((D)/(2))^(2)xx100=25pi D^(2)`
Now, increase in diameter = 20% of D
`=Dxx(20)/(100)=(D)/(5)`
and new diameter `=D+(D)/(5)=(6D)/(5)`
`rArr` new radius `=(6D)/(5xx2)=(3D)/(5)`
Let new height = h
`therefore` Volume `=pi ((3D)/(5))^(2) h = (9)/(25)pi D^(2)h`
Given that, `(9)/(25)pi D^(2)h=25 pi D^(2)`
`rArr h = (625)/(9)`
`therefore` decrease in height `=100-(625)/(9)=(900-625)/(9)=(275)/(9)`
and % decrease in height`=(275)/(9xx100)xx100%=30(5)/(9)%`
Alternative Method (Short Trick) :
`V=pi r^(2)h i.e., r,r,h` (`pi` is constant)
% change in volume `-a+b+c+(ab+bc+ca)/(100)+(abc)/((100)^(2))`
Here `0=20+20-h+(20(20)+20(-h)+(-h)(20))/(100)+(20(20)(-h))/(100xx100)`
`rArr 0=40-h+(400-40h)/(100)-(4h)/(100)`
`rArr h=40=(400-44h)/(100)`
`rArr 100h-4000=400-44h`
`rArr 144h=4400 rArr h=(4400)/(144)=30(5)/(9)%`