Question

A hollow garden roller is 63 cm wide with a grit of 440 cm, and is made of 4cm thick iron. Find the volume of iron used.

Answer 1

Hint: A cylinder is a three-dimensional solid shape of geometry which has two circular bases at a fixed distance and a curved surface. A right circular cylinder has a surface perpendicular to its circular bases and it is a perfectly symmetrical shape.
The volume of a shape is generally referred to as the capacity it can hold. The volume of a cylinder determines the capacity cylinder has. The volume of the cylinder is proportional to its diameter and its length. The larger will be the quantities larger will be its capacity.


Complete step by step solution:
Height of roller, h=63cm
Outer Grit of the roller is the circumference or, perimeter of the roller 440cm, hence the radius of the roller will be:
\[
  2\pi r = 440 \\
  r = \dfrac{{440}}{{2\pi }} \\
   = \dfrac{{440}}{{2 \times \dfrac{{22}}{7}}} \\
   = \dfrac{{440 \times 7}}{{2 \times 22}} \\
   = 70cm \\
 \]
It is mentioned that the roller is a hollow roller and the thickness is 4cm hence we have to find the radius for the hollow part
\[
  r' = r - 4 \\
   = 70 - 4 \\
   = 66cm \\
 \]
Since the cylinder is hollow we have to find the volume of iron for roller thick part \[\left( {r - r'} \right)\],
Hence, the volume of iron will be:
\[
  V = \pi \left( {{r^2} - {{\left( {r'} \right)}^2}} \right)h \\
   = \dfrac{{22}}{7}\left( {{{\left( {70} \right)}^2} - {{\left( {66} \right)}^2}} \right) \times 63 \\
   = \dfrac{{22}}{7}\left( {70 + 66} \right)\left( {70 - 66} \right) \times 63 \\
   = 22 \times 136 \times 4 \times 9 \\
   = 107712{\text{ }}c{m^3} \\
 \]
The volume of a 4cm thick iron cylinder is \[107712{\text{ }}c{m^3}\].

Additional Information: If the function is given as the difference of the square of the two different numbers then, \[\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right)\] holds true.

Note: In the case of hollow objects do not find the volume for the hollow part of the objects. It is always confusing for the hollow cylinder as the thickness of the material by which the body has been made is given instead of the inner radius of the body, so be careful while using the inner radius of the body, it the difference of the outer radius and the thickness of the material by which the body has been made.