Question

How many balls can be placed in the cylindrical tube if they fit exactly as shown in the figure. The inner curved surface area of the cylinder is $30.8m^2$ and the surface area of each ball is $61600c{m}^2$. (Use $=\frac{22}{7}$)

Answer 1

Given,
Inner curved surface area $=30.8m^2=30.8\times {10}^4=308000c{m}^2$

Surface area of 1 ball $=61600c{m}^2$

The number of balls that can be placed inside the cylinder can be found from the ratio between the length of the cylinder and the diameter of the balls.

To find the diameter of the balls:
It is given, surface area of a ball,
$\Rightarrow 4\pi r^2=61600c{m}^2$
$\Rightarrow r^2=\frac{61600}{4}\times \frac{7}{22}$
$\Rightarrow r^2=15400\times \frac{7}{22}$
$\Rightarrow r^2=700\times 7$
$r=\sqrt{4900}$
$r=70cm$

So,
the diameter of a ball (2r) is $140cm$.

To find the length of the cylinder:
It is given, the inner curved surface area $2\pi rl=308000c{m}^2$

(Since the balls fit exactly into the tube the radius of the balls and the radius of the inner surface of the cylinder are the same.)
$\Rightarrow 2\pi \times 70\times l=308000c{m}^2$
$\Rightarrow l=308000\times \frac{7}{22}\times \frac{1}{2}\times \frac{1}{70}$
$\Rightarrow l=154000\times \frac{7}{22}\times \frac{1}{70}$
$\Rightarrow l=700cm$

The length of the cylinder is $700cm$
Number of balls that can be placed in the cylinder
$=\frac{Lengthofthecylinder}{Diameterofeachball}$
$=\frac{700}{140}$
$=5$

So, the number of balls that can be placed in the cylinder is 5.