A) \[269\frac{1}{2}\,\,{{\text{m}}^{3}}\]
B) \[259\frac{1}{2}\,\,{{\text{m}}^{3}}\]
C) \[249\frac{1}{2}\,\,{{\text{m}}^{3}}\]
D) \[239\frac{1}{2}\,\,{{\text{m}}^{3}}\]
Correct Answer: A
Solution :
| Given, total surface area \[=231\,\,{{\text{m}}^{2}}\] |
| and curved surface area \[=\frac{2}{3}\times \]total surface area |
| \[=\frac{2}{3}\times 231=154\,\,{{\text{m}}^{2}}\] |
| Now, \[\frac{\text{Total}\,\,\text{surface}\,\,\text{area}}{\text{Curved}\,\,\text{surface}\,\,\text{area}}=\frac{231}{154}\] |
| \[\Rightarrow \] \[\frac{2\pi rh+2\pi {{r}^{2}}}{2\pi rh}=\frac{3}{2}\] |
| \[\Rightarrow \] \[\frac{h+r}{h}=\frac{3}{2}\] |
| \[\Rightarrow \] \[2h+2r=3h\] |
| \[\Rightarrow \] \[2r=3h-2h\] |
| \[\therefore \] \[h=2r\] |
| \[\because \] Curved surface area \[=2\pi rh=154\] |
| \[\Rightarrow \] \[2\pi \,\,(2r)=154\]\[\Rightarrow \]\[r=\frac{7}{2}\text{m}\]\[\therefore \]\[h=7\,\,\text{m}\] |
| \[\therefore \] Required volume \[=\pi {{r}^{2}}h\] |
| \[=\frac{22}{7}\times \frac{7}{2}\times \frac{7}{2}\times 7=\frac{539}{2}\] |
| \[=269\frac{1}{2}{{\text{m}}^{3}}\] |
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