Question

An iron spherical ball has been melted and recast into smaller balls of equal size. If the radius of each of the smaller balls is 1/4 of the radius of the original ball, how many such balls are made? Compare the surface area, of all the smaller balls combined together with that of the original ball.

Answer 1

Let the radius of the big metallic ball is 4r. Therefore, the volume of the big metallic ball is

The metallic sphere is melted to produce small balls of radius. Then, the volume of each of the small balls is

Since, the volume of the big metallic ball is equal to the sum of the volumes of the small balls, we have the number of produced small balls is

Hence, the number of small balls is

The surface area of the big ball is

The surface area of each of the small ball is

Therefore, the total surface area of the 64 small balls is

Now, we compute the following ratio

Hence, the total surface area of the small balls is equal to four times the surface area of the original big ball.