Question

A lead pencil is in the shape of a right circular cylinder. The pencil is 28cm long and its radius is 3mm. If the lead is of radius 1mm, then find the volume of the wood in the pencil.
A. \[7.04\;c{m^3}\]
B. \[7.14\;c{m^3}\]
C. \[8.04\;c{m^3}\]
D.None of these

Answer 1

Hint: The volume of a hollow cylinder is \[V = \pi \left( {{R^2} - {r^2}} \right)h\] , where \[R\] is the radius of outer surface of hollow cylinder, \[r\] is the radius of inner surface of hollow cylinder and \[h\] is the height of the cylinder.
In this question the radius of the pencil and the radius of the lead is given which becomes the inner and outer radius of the cylinder and also the height of the pencil is given so by substituting these values we will find the volume of the wood in the pencil

Complete step-by-step answer:
Given
The height of the pencil \[h = 28\;cm\]
Radius of the pencil \[R = 3mm = 0.3\;cm\]
Radius of the lead \[r = 1mm = 0.1\;cm\]
We need to find the volume of the wood in the pencil. This means if we find the volume of the wood of the pencil we need to remove the lead of the pencil and if we remove the lead off the pencil then we will be left with the hollow wooden pencil whose volume is needed.
We know the volume of a hollow cylinder is given by the formula \[V = \pi \left( {{R^2} - {r^2}} \right)h\]
Now we substitute the given data from the question in the above formula, hence we get
  \[
  V = \pi \left( {{R^2} - {r^2}} \right)h \\
   = \pi \left( {{{\left( {0.3} \right)}^2} - {{\left( {0.1} \right)}^2}} \right) \times \left( {28} \right) \]
Now we further solve the equation by substituting \[\pi = \dfrac{{22}}{7}\] in the formula, we get
 \[
  V = \dfrac{{22}}{7} \times \left( {{{\left( {0.3} \right)}^2} - {{\left( {0.1} \right)}^2}} \right) \times \left( {28} \right) \\
   = \dfrac{{22}}{7} \times \left( {0.09 - 0.01} \right) \times 28 \\
   = 22 \times 0.08 \times 4 \\
   = 88 \times 0.08 \\
   = 7.04\;c{m^2} \;
 \]
Hence we get the value of \[V = 7.04\;c{m^2}\]
Therefore the volume of the wood in the pencil is \[ = 7.04\;c{m^2}\]
So, the correct answer is “Option A”.

Note: Another method that can be used to solve this question is to find the volume of pencil considering the lead to it and also finding the volume of the lead separately and then subtracting the volume of lead to the volume of the pencil to find the remaining volume which is the wooden part of the pencil.
 \[V = \pi {R^2}h - \pi {r^2}h\]