Question

A cylindrical road roller made of iron is 1 m long. Its inner diameter is 54 cm and the thickness of the iron sheet rolled into the road roller is 9cm. Find the weight of the roller if 1$cm^{3}$ of iron weights 8g.
A) 45g
B) 24g
C) 36g
D) None of the above.

Answer 1

Hint: In this question it is given that a cylindrical road roller made of iron is 1 m long. Its inner diameter is 54 cm and the thickness of the iron sheet rolled into the road roller is 9cm. We have to find the weight of the roller if 1$cm^{3}$ of iron weighs 8g. So to find the solution we have to find the volume of a hollow thick cylinder. So the volume is, V=$$\pi \left( r^{2}_{2}-r^{2}_{1}\right) h$$................(1)
Where $$r_{1}\ and\ r_{2}$$ is the inner and outer radius and h is the height of the cylinder.

Complete step-by-step solution:
It is given that the road roller is 1 m long.
That implies the height of the cylinder, h=1m=100cm.
Inner diameter =54cm, which implies the inner radius,
$r_{1}=\dfrac{54}{2}$cm=27cm.
Since the thickness is given 9cm, then the outer radius will be,
$r_{2}=\left( r_{1}+\text{thickness}\right) $=(27+9) cm=36cm.
Then from equation (1) we can say that the volume of the roller,
V=$$\pi \left( r^{2}_{2}-r^{2}_{1}\right) h$$
  =$$\pi \left( 36^{2}-27^{2}\right) \times 100\ cm^{3}$$
  =$$\dfrac{22}{7} \times \left( 1296-729\right) \times 100\ cm^{3}$$
  =$$\dfrac{22}{7} \times 567\times 100\ cm^{3}$$
  =178200 $cm^{3}$

Now we have to find the weight of the roller,
So it is given that,
 1$cm^{3}$ of iron weights 8g
$\therefore 178200cm^{3}$ of iron weights=(8$\times$178200)g=1425600g.
Thus the weight of the roller is 1425600g
Hence the correct option is option D.
Note: To solve this question you need to know that for a hollow thick cylinder the outer radius is the summation of inner radius and thickness. Also you can find the volume of the hollow cylinder just by subtracting the volume of the inner cylinder from the outer cylinder, i.e, $\pi r^{2}_{2}h-\pi r^{2}_{1}h$, which will give us the formula for hollow cylinder i.e, Volume(V)=$$\pi \left( r^{2}_{2}-r^{2}_{1}\right) h$$.