The ratio between the radius of the base and the height of a cylinder is 2 : 3. If its volume is 1617 cm3, the total surface area of the cylinder is
(a) 308 cm2
(b) 462 cm2
(c) 540 cm2
(d) 770 cm2
(d) 770 cm2
Let the common multiple be x.
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,
$\frac{22}{7} \times(2 x)^{2} \times 3 x=1617$
$\Rightarrow \frac{22}{7} \times 4 x^{2} \times 3 x=1617$
$\Rightarrow \frac{22}{7} \times 12 x^{3}=1617$
$\Rightarrow x^{3}=\left(1617 \times \frac{7}{22} \times \frac{1}{12}\right)$
$\Rightarrow x^{3}=\left(\frac{7}{2} \times \frac{7}{2} \times \frac{7}{2}\right)$
$\Rightarrow x^{3}=\left(\frac{7}{2}\right)^{3}$
$\Rightarrow x=\frac{7}{2}$
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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