Question

Height of a solid cylinder is 10 cm and diameter 8 cm. Two equal conical holes have been made from its both ends of diameter of the hole is 6 cm and height is 4 cm. Find the volume of the cylinder, the volume of one conical hole and the volume of remaining solid.

Answer 1

Hint: To solve the question, we have to apply the formula for volume of cylinder and volume of cone to calculate the answer. To answer the last question, we have to analyse that the volume of the remaining solid is equal to difference of the volume of the solid cylinder and the volume of the two conical holes.

Complete step-by-step answer:

The given height of a solid cylinder is equal to 10 cm.
The given diameter of a solid cylinder is equal to 8 cm.
We know the formula for the volume of the cylinder is given by \[\pi {{r}^{2}}h\]
Where h, r are the height and radius of the cylinder respectively.
We know diameter = 2(radius). Thus, we get
\[\begin{align}
  & 8=2r \\
 & r=\dfrac{8}{2}=4cm \\
\end{align}\]
By substituting the given values in the above formula, we get
\[\begin{align}
  & \pi {{(4)}^{2}}10 \\
 & =\pi \times 16\times 10 \\
 & =160\pi \\
\end{align}\]
Thus, the volume of the solid cylinder is equal to \[160\pi \] cubic cm.

The given height of a conical hole is equal to 4 cm.
The given diameter of a conical hole is equal to 6 cm.
We know the formula for the volume of the cone is given by \[\dfrac{1}{3}\pi {{r}^{2}}h\]
Where h, r are the height and radius of the cone respectively.
We know diameter = 2(radius). Thus, we get
\[\begin{align}
  & 6=2r \\
 & r=\dfrac{6}{2}=3cm \\
\end{align}\]
By substituting the given values in the above formula, we get
\[\begin{align}
  & \dfrac{1}{3}\pi {{(3)}^{2}}4 \\
 & =\pi (3)4 \\
 & =12\pi \\
\end{align}\]
Thus, the volume of one conical hole is equal to \[12\pi \] cubic cm.
The volume of the remaining solid = The volume of the solid cylinder –the volume of the conical holes
Since, there are two conical holes, we get
The volume of the remaining solid = The volume of the solid cylinder – 2(the volume of one conical hole)
By substituting the given values in the above formula, we get
\[\begin{align}
  & =160\pi -2(12\pi ) \\
 & =160\pi -24\pi \\
 & =136\pi \\
\end{align}\]
Thus, the volume of the remaining solid is equal to \[136\pi \]

Note: The possibility of mistake can be, not applying the formula for volume of cylinder and volume of cone to calculate the answer. The other possibility of mistake can be, not analysing that the volume of the remaining solid is equal to difference of the volume of the solid cylinder and the volume of the two conical holes.