Question

The number of balls of radius 1 cm that can be made from a solid sphere of radius 4 cm is
A.64
B.16
C.12
D.4

Answer 1

Hint: First of all we need to calculate the volume of both the spheres are of radius 1 cm and radius 4 cm. Using the formula of volume \[v = \dfrac{{4\pi }}{3}{r^3}\] . Hence, on calculating the volumes of both spheres and on taking the ratio of bigger solid sphere to smaller solid sphere hence, total number of balls can be obtained.

Complete step-by-step answer:
As the given information, the radius of the spheres are of radius 1 cm and radius 4 cm.
Diagram:

Volume of a sphere is given by \[v = \dfrac{{4\pi }}{3}{r^3}\]
Let \[{r_1}\]is radius of bigger sphere so \[{r_1} = 4\,cm\]
And \[{r_2}\]is radius of smaller sphere so \[{r_2} = 1c.m\]
Hence, now for calculating the numbers of balls we take the ratios of volume of the bigger sphere to the smaller sphere.
\[n = \dfrac{{\dfrac{{4\pi }}{3}{r_1}^3}}{{\dfrac{{4\pi }}{3}{r_2}^3}}\]
On cancelling out common terms, we get,
\[n = \dfrac{{{r_1}^3}}{{{r_2}^3}}\]
Now, substituting the values of radius, \[{r_1} = 4cm\] and \[{r_2} = 1cm\], we get,
\[n = \dfrac{{{4^3}}}{{{1^3}}}\]
On simplification we get,

\[n = 64\]
Hence, the number of balls of radius 1 cm that can be made from a solid sphere of radius 4 cm is \[64\].
Hence, option (A) is the correct answer.

Note: You can think that if we scoop out spheres from a big sphere there would be waste portion, and hence the number of balls can be less than 64, but here we are not considering that case, and we will use the extra part and reform it to make balls.
A sphere is a geometrical object in three-dimensional space that is the surface of a ball. Like a circle in a two-dimensional space, a sphere is defined mathematically as the set of points that are all at the same distance r from a given point in a three-dimensional space.