| The radius of the base of a right circular cone is increased by 15% keeping the height fixed. The volume of the cone will be increased by |
A) 30%
B) 31%
C) 32.25%
D) 34.75%
Correct Answer: C
Solution :
| Let the fixed height of a right circular cone be h and initial radius be r. |
| Then, initial volume of cone,\[{{V}_{1}}=\frac{1}{3}\pi {{r}^{2}}h\] |
| After increasing 15% radius of a cone |
| \[=\left( r+\frac{3r}{20} \right)=\frac{23}{20}r\] |
| New volume becomes, |
| \[{{V}_{2}}=\frac{1}{3}\pi {{\left( \frac{23}{20} \right)}^{2}}{{r}^{2}}h\] |
| \[\therefore \]Increasing percentage \[=\left( \frac{{{V}_{2}}-{{V}_{1}}}{{{V}_{1}}} \right)\times 100\] |
| \[=\frac{\frac{1}{3}\pi {{r}^{2}}h}{\frac{1}{3}\pi {{r}^{2}}h}\left\{ {{\left( \frac{23}{20} \right)}^{2}}-1 \right\}\times 100\] |
| \[=\left( \frac{23}{20}+1 \right)\left( \frac{23}{20}-1 \right)\times 100\] |
| \[=\frac{43}{20}\times \frac{3}{20}\times 100=32.25\]% |
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