Question

Find the volume of the largest right circular cone that can be cut out from a cube of edge 4.2cm.

Answer 1

Hint: For solving this question, the edge of the cube is given. So, for the largest volume, the diameter of the largest circular cone is the same as the edge of the cube. As we know that radius is half of diameter, using this we get the radius of the largest circular cone. Similarly, the height of the cone is also equal to the edge of the cube. Using the formula volume of the cone, we get the volume of the largest circular cone.

Complete step-by-step solution -

Formula of volume of cone used in the question is:$V=\dfrac{1}{3}\pi {{r}^{2}}h$.
Here, r is the radius of the cone and h is the height of the cone.
According to the given problem statement, the edge of cube = 4.2 cm
Diameter of the largest circular cone that can be of shape cone = 4.2 cm
Radius of the largest circular is half of the diameter of the largest circular cone.
$\begin{align}
  & r=\dfrac{d}{2} \\
 & r=\dfrac{4.2}{2} \\
 & r=2.1\text{cm} \\
\end{align}$
Height of the largest circular cone = h = 4.2 cm

Volume of the largest circular cone $=\dfrac{1}{3}\pi {{r}^{2}}h$.
Volume of the largest circular cone $=\dfrac{1}{3}\times \dfrac{22}{7}\times 2.1\times 2.1\times 4.2c{{m}^{3}}$
Volume of the largest circular cone$=19.40\text{c}{{\text{m}}^{3}}$.
Hence, the volume of the largest circular cone is $19.40c{{m}^{3}}$.
Note: The key concept involved in solving this problem is the knowledge of the concept of maximum possible volume for a right circular cone. Two major steps for solving the problem is the assumption of height and diameter of the cone to be of the same dimension as that of the cube side.