Question

The largest possible sphere is carried out a wooden solid cube of side 7cm. Find the volume of the wood left. $ [\pi = \dfrac{{22}}{7}] $
A. 163.3
B. 164
C. 165
D. 170

Answer 1

Hint: To solve the above question, first we will obtain the volume of the cube from the side of the cube. Then from the given data we will derive diameter, thus by the radius of the sphere. From that we will obtain the volume of the sphere. By subtracting the volume of the sphere from the volume of the cube we will get the volume of the wooden cube left.

Complete step-by-step answer:
According to the question, the largest possible sphere is carried out in a wooden solid cube.
It is given that the side of the wooden solid cube is $ a = 7 $ cm
Hence, the volume of the cube is, $ {a^3} = {7^3} = 343c{m^3} $
The largest possible diameter of the sphere = side of the cube = 7cm
 $ \therefore $ The radius of the sphere, $ r = \dfrac{7}{2}cm $
 $ \therefore $ The volume of the sphere is, $ \dfrac{4}{3}\pi {r^3} = \dfrac{4}{3} \times \dfrac{{22}}{7} \times {\left( {\dfrac{7}{2}} \right)^3} $
The volume of the sphere is $ = \dfrac{{539}}{3}c{m^3} $
Hence, the volume of wood left = volume of the cube – volume of the sphere
 $ = 343 - \dfrac{{539}}{3} $
 $ = \dfrac{{1029 - 539}}{3} = \dfrac{{490}}{3} $
 $ = 163.3c{m^3} $
 $ \therefore $ The volume of the wood left is $ 163.3c{m^3} $

So, the correct answer is “Option A”.

Note: A cube is a three dimensional solid object bounded by 6 square faces. It has 6 faces, 12 edges and 8 vertices.
The area of the cube is given by, $ 6{a^2} $ , where a is the side of the cube.
The volume of the cube is given by, $ {a^3} $ .
A sphere is a three dimensional sphere defined as a set of points from the given point called center with an equal distance in the three dimensional space.
The volume of the sphere is given by, $ \dfrac{4}{3}\pi {r^3} $ , where r is the radius of the sphere.
You should remember all the formulas of three dimensional geometrical shapes.