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

The diameter of a right circular cylinder is increased by 20%. Find th

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