The radius of a solid right circular cylinder decreases by 20% and its height increases by 10%. Find the percentage change in its: (i) volume (ii) curved surface area
The radius of a solid right circular cylinder decreases by 20% and its height increases by 10%. Find the percentage change in its: (i) volume (ii) curved surface area

Let the original dimensions of the solid right cylinder be radius (r) and height (h) in cm.

Then its volume = πr2h cm3and curved surface area = 2πrh

Now, after the changes the new dimensions are:

New radius (r’) = r – 0.2r = 0.8r and

New height (h’) = h + 0.1h = 1.1h

The new volume

    \[\begin{array}{*{35}{l}} =\text{ }\pi r{{}^{2}}h\text{ }c{{m}^{3}}  \\ =\text{ }\pi {{\left( 0.8r \right)}^{2}}\left( 1.1h \right)\text{ }c{{m}^{3}}  \\ =\text{ }0.704\text{ }\pi {{r}^{2}}h\text{ }c{{m}^{3}}  \\ \end{array}\]

And, the new curved surface area

    \[\begin{array}{*{35}{l}} =\text{ }2\pi rh\text{ }=\text{ }2\pi \left( 0.8r \right)\left( 1.1h \right)  \\ =\text{ }\left( 0.88 \right)\text{ }2\pi rh  \\ \end{array}\]

(i) Percentage change in its volume = Decrease in volume/ original volume x 100 %

= (Original volume – new volume)/ original volume x 100 %

    \[\begin{array}{*{35}{l}} =\text{ }\left( \pi {{r}^{2}}h\text{ }-\text{ }0.704\text{ }\pi {{r}^{2}}h \right)/\text{ }\pi {{r}^{2}}h\text{ x }100  \\ =\text{ }0.296\text{ x }100\text{ }=\text{ }29.6\text{ }%  \\ \end{array}\]

(ii) Percentage change in its curved surface area = Decrease in CSA/ original CSA x 100 %

= (Original CSA – new CSA)/ original CSA x 100 %

    \[\begin{array}{*{35}{l}} =\text{ }\left( 2\pi rh\text{ }\text{ }-\left( 0.88 \right)\text{ }2\pi rh \right)/\text{ }2\pi rh\text{ x }100  \\ =\text{ }0.12\text{ }x\text{ }100\text{ }=\text{ }12\text{ }%  \\ \end{array}\]