question_answer

Radius of a cylinder is r and the height is h. Find the change in the volume if the

(a) height is doubled

(b) height is doubled and the radius is halved

(c) height remains same and the radius is halved.

Answer:

Volume of cylinder \[=\pi {{r}^{2}}h\]

(a) Height is doubled i.e., h' =2h

Volume of cylinder \[=\pi {{r}^{2}}h'\]

\[=\pi {{r}^{2}}(2h)\]

\[=2\pi {{r}^{2}}h\]     (Double of the original)

(b) \[h'=2h\]and\[r'=\frac{r}{2}\]

Then volume of cylinder \[=\pi {{r}^{2}}h\]

\[=\pi {{\left( \frac{r}{2} \right)}^{2}}\times 2h\]

\[=\pi \,\times \frac{{{r}^{2}}}{4}\times 2h\]

\[=\frac{1}{2}\pi {{r}^{2}}h\]          (Half of the original)

(c)       \[r'=\frac{r}{2}\] unit

Volume of cylinder \[=\pi r{{'}^{2}}h\]

\[=\pi \,{{\left( \frac{r}{2} \right)}^{2}}h\]

\[=\frac{1}{4}\pi {{r}^{2}}h\]cubic unit (One fourth of the original)