Question Video: Finding the Radii of Equal Spheres Formed from a Bigger One of Known Radius Mathematics • Second Year of Preparatory School

Video Transcript

A sphere of metal with radius 14.1 centimeters was melted down and formed into four equal spheres. Find the radius of one of the smaller spheres, giving your answer to the nearest centimeter.

In this example, we have a single solid sphere which is melted down and reformed into four equally sized smaller spheres. The larger sphere has a radius of 14.1 centimeters. Our first step is to work out the volume of metal contained in the original sphere using the formula for the volume of a sphere. That’s four over three 𝜋𝑟 cubed, where 𝑟 is the sphere’s radius. We can then divide our result into four equal parts and use the formula in reverse to find the radius of the four identical smaller spheres.

Calling our original sphere 𝑆 one and the smaller spheres 𝑆 four, the volume of 𝑆 one, that’s 𝑉 one, equals four over three 𝜋 times 14.1 cubed. 14.1 cubed is 2803.221. And so the volume of our original sphere is four over three 𝜋 times 2803.221 cubic centimeters.

Now, we could work this out. However, remember, to find the volume of each of the four equal smaller spheres, we want to divide this by four. And since it contains a factor of four, we might just as well leave it as it is. So the volume 𝑉 four of each of the smaller spheres is 𝜋 by three times 2803.221.

We know from our formula that this is equal to four over three 𝜋𝑟 cubed. And it’s the radius 𝑟 that we want to find. So, now multiplying both sides by three and dividing through by 𝜋, we have 2803.221 equals four 𝑟 cubed. And then dividing through by four, we’re left with 𝑟 cubed equal to 2803.221 over four. Now, taking the cube root on both sides, we have the radius equal to 8.882 and so on. Hence, to the nearest centimeter, the radius of one of the smaller spheres is equal to nine centimeters.