The radius of the base and the height of a solid right circular cylinder are in the ratio 2 : 3 and its volume is 1617 cm3.
The radius of the base and the height of a solid right circular cylinder are in the ratio 2 : 3 and its volume is 1617 cm3. Find the total surface area of the cylinder.
Let the radius of the cylinder be 2x cm and its height be 3x cm.
Then, volume of the cylinder $=\pi r^{2} h$
$=\frac{22}{7} \times(2 x)^{2} \times 3 x$
Therefore,
$\frac{22}{7} \times(2 x)^{2} \times 3 x=1617$
$\Rightarrow \frac{22}{7} \times 4 x^{2} \times 3 x=1617$
$\Rightarrow \frac{22}{7} \times 12 x^{3}=1617$
$\Rightarrow x^{3}=\left(1617 \times \frac{7}{22} \times 12\right)$
$\Rightarrow x^{3}=\left(\frac{7}{2} \times \frac{7}{2} \times \frac{7}{2}\right)$
$\Rightarrow x^{3}=\left(\frac{7}{2}\right)^{3}$
$\Rightarrow x=\frac{7}{2}$
Now, $r=7 \mathrm{~cm}$ and $h=\frac{21}{2} \mathrm{~cm}$
Hence, the total surface area of the cylinder:
$\left(2 \pi r h+2 \pi r^{2}\right)$
$=2 \pi r(h+r)$
$=2 \times \frac{22}{7} \times 7 \times\left(\frac{21}{2}+7\right) \mathrm{cm}^{2}$
$=\left(2 \times \frac{22}{7} \times 7 \times \frac{35}{2}\right) \mathrm{cm}^{2}$
$=770 \mathrm{~cm}^{2}$
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