| A cylinder and a cone have equal radii of their bases and equal heights. If their curved surface areas are in the ratio 8 : 5, then the ratio of their radius and height is [SSC (CGL) 2011] |
A) 1: 2
B) 1 : 3
C) 2 : 3
D) 3 : 4
Correct Answer: D
Solution :
| Radius of cylinder = Radius of cone = r |
| Height of cylinder = Height of cone = h |
| \[\therefore \]\[\frac{\text{Surface}\,\,\text{area}\,\,\text{of}\,\,\text{cylinder}}{\text{Surface}\,\,\text{area}\,\,\text{of}\,\,\text{cone}}=\frac{2\pi rh}{\pi rl}=\frac{8}{5}\]\[\Rightarrow \]\[2h=\frac{8l}{5}\] |
| Now, \[2h=\frac{8l}{5}\] |
| On squaring both sides' we get |
| \[4{{h}^{2}}=\frac{{{8}^{2}}{{l}^{2}}}{{{5}^{2}}}\]\[\Rightarrow \]\[4{{h}^{2}}=\frac{{{8}^{2}}}{{{5}^{2}}}({{h}^{2}}+{{r}^{2}})\] |
| \[\Rightarrow \] \[4{{h}^{2}}-\frac{64{{h}^{2}}}{25}=\frac{{{8}^{2}}}{{{5}^{2}}}{{r}^{2}}\] |
| \[\Rightarrow \] \[\frac{(100-64){{h}^{2}}}{25}=\frac{64}{25}{{r}^{2}}\] |
| \[\Rightarrow \] \[\frac{{{r}^{2}}}{{{h}^{2}}}=\frac{36}{64}\]\[\Rightarrow \]\[\frac{r}{h}=\frac{6}{8}=\frac{3}{4}\] |
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