Question : The radius of the cylindrical milk container is half its height, and the surface area of the inner part is 616 sq cm. The amount of milk that the container can hold, approximately, is _____. Use : $\sqrt{5 } =2.23$ and $\pi=\frac{22}{7}$
Option 1: 1.42 litres
Option 2: 1.53 litres
Option 3: 1.71 litres
Option 4: 1.82 litres
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Correct Answer: 1.53 litres
Solution :
Let $r$ be the radius and $h$ be the height of the cylinder.
Given: $r = \frac{h}{2}$
The total surface area of the open cylinder = 616 sq cm.
We know that,
The total surface area of the open cylinder $=2\pi r h+\pi r^2$
$⇒ 616 = 2\pi r h+\pi r^2$
$⇒ 616 = \pi r(2h+r)$
Putting $r = \frac{h}{2}$, we get:
$⇒616 = \pi\times\frac{h}{2}(2h+\frac{h}{2})$
$⇒ 616 = \pi\times\frac{5h^2}{4}$
$⇒ h^2= \frac{616\times7\times4}{5\times22}$
$\therefore h=\frac{28}{\sqrt5}$
So, $r=\frac{h}{2}=\frac{\frac{28}{\sqrt5}}{2}=\frac{14}{\sqrt5}$
The volume of the cylinder
$=\pi r^2 h$
$=\frac{22}{7}×(\frac{14}{\sqrt5})^2×\frac{28}{\sqrt5}$
$=\frac{22}{7}×\frac{196}{5}×\frac{28}{2.23}$
$=1546.9$ cm
3
$\approx 1.53$ litres
Hence, the correct answer is 1.53 litres.
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