Question

The largest possible sphere is carved out of a wooden solid cube of side 7 cm. Find the volume of the wood left. (Use \[\pi =3.14\]).

Answer 1

Hint: In this question, we first need to find the radius of the sphere by equation the diameter of the sphere to the side of the cube. Then we can get the volume of the wood left from the volume of the sphere and the cube.
Volume of the cube is given by \[{{s}^{3}}\]
Volume of the sphere is given by \[\dfrac{4}{3}\pi {{r}^{3}}\]

Complete step by step solution:
CUBOID: A figure which is surrounded by six rectangular surfaces is called CUBOID.
CUBE: A CUBOID whose length, breadth and height are the same is called a cube.
Let the side of the cube be s.
Total surface area is \[6{{s}^{2}}\]
Volume of the cube is \[{{s}^{3}}\]
SPHERE: A sphere is a solid generated by the revolution of a semicircle about its diameter.
Surface area of the sphere is given by \[4\pi {{r}^{2}}\]
Volume of the sphere is given by \[\dfrac{4}{3}\pi {{r}^{3}}\]

Where, r is the radius of the sphere.
Given in the question that side of the cube is 7 cm.
\[s=7\]
As we know that diameter is the longest side in a sphere. Here, the side of the cube will be the diameter of the sphere.
Now, by equating the diameter of the sphere with side of the cube we get,
\[\begin{align}
  & \Rightarrow 2r=s \\
 & \Rightarrow r=\dfrac{s}{2} \\
 & \therefore r=\dfrac{7}{2} \\
\end{align}\]
Now, the volume of the wood left can be calculated by subtracting the volume of the sphere from the volume of the cube.
Let us assume that the volume of wood left as W, volume of the sphere as S and volume of the cube as C.
\[\begin{align}
  & \Rightarrow W=C-S \\
 & \Rightarrow W={{s}^{3}}-\dfrac{4}{3}\pi {{r}^{3}} \\
\end{align}\]
Now, by substituting the corresponding values in the above equation we get,
\[\begin{align}
  & \Rightarrow W={{7}^{3}}-\dfrac{4}{3}\pi {{\left( \dfrac{7}{2} \right)}^{3}}\text{ }\left[ \because s=7,r=\dfrac{7}{2} \right] \\
 & \Rightarrow W={{7}^{3}}\left( 1-\dfrac{4}{3}\times \pi \times \dfrac{1}{8} \right) \\
\end{align}\]
\[\Rightarrow W={{7}^{3}}\left( 1-3.14\times \dfrac{1}{6} \right)\text{ }\left[ \because \pi =3.14 \right]\]
\[\Rightarrow W={{7}^{3}}\times \dfrac{2.86}{6}\]
\[\begin{align}
  & \Rightarrow W={{7}^{3}}\times 0.477 \\
 & \therefore W=163.611c{{m}^{3}} \\
\end{align}\]
Hence, the volume of the wood left is 163.611 cubic centimetre.

Note: Instead of assuming different variables for the volume of the sphere, volume of the cube and volume of the wood left we can directly assume one variable to the volume of the wood left and then equate it to the difference in the volumes of sphere and cube. Both the methods give the same result just that the number of steps will be less.
It is important to note that the volume of the wood left is the difference between the volume of the cube and the volume of the sphere. Because here we are carving out a sphere from the solid cube which means that we are removing the sphere from the cube.
Here to find the radius of the sphere we equated the diameter of the sphere to the side of the cube because the diameter is the longest possible length that can be removed which will be equal to the side of the cube.