Question

Number if balls, each diameter 1cm that can be made from a sphere if diameter 12 cm is?

Answer 1

Hey there !

Solution:

Let the number of balls be 'n'

Volume of sphere obtained by merging 'n' balls can be calculated as:

$\text{Volume of a Sphere} = \dfrac{4}{3} \pi r^3 \\ \\ \text{ Volume of Sphere obtained by combining 'n' balls} = n \times \dfrac{4}{3} \pi r^3 \\ \\ \text{ It is given that the diameter of the sphere obtained is 12 cm.} \\ \\ \implies \text{Radius} = \dfrac{ Diameter }{2} = \dfrac{12}{2} = 6 \: cm$

$\text{ Therefore substituting radius in the formula we get,} \\ \\ \implies Volume \times n = \dfrac{4}{3} \pi \ (6)^3 \\ \\ \implies Volume \times n = \dfrac{4 \times 216 \times \pi}{3} \\ \\ \implies Volume \times n = \dfrac{864}{3} \pi \\ \\ \implies Volume \times n = 288 \pi \\ \\ \\ \text{ Hence Volume of the sphere made out of n balls is}\: \dfrac{288 \pi}{n}$

$\text{ Now let us find the value of 1 ball} \\ \\ \text{ Diameter of one ball is 1 cm. Therefore the radius is:} \\ \\ \implies Radius = \dfrac{Diameter}{2} = \dfrac{1}{2} = 0.5cm \\ \\ \text{ Therefore Volume of one ball = } \\ \\ \implies Volume\:of\:ball = \dfrac{4}{3} \pi \ ( 0.5 )^3 = \dfrac{4 \times 0.125 \times \pi}{3} = \dfrac{0.5 \pi}{3}$

$\text{Therefore,} \\ \\ \\ \implies \text{ Volume of sphere made by n balls } = \text{ Volume of one ball} \: \times n \\ \\ \\ \implies 288\: \pi = \dfrac{0.5 \pi}{3} \times n \\ \\ \\ \implies n = \dfrac{ 288\: \pi \times 3}{0.5 \: \pi} \\ \\ \\ \implies n = 576 \times 3 = 1728 \: balls$

$\boxed{ Hence\: the\: number\: of\: balls\: is\: 1728.}$

Hope my answer helped !