4 smaller spheres of equal radius are created using the material used to form a larger sphere. How does the volume change in the two cases.

Expert Answers
justaguide eNotes educator| Certified Educator

The surface area of a sphere with radius r is `(4/3)*pi*r^3` and the surface area of the sphere is `4*pi*r^2` .

Let the radius of the larger sphere be R, the surface area is `4*pi*R^2` , as the same material is used to create four smaller spheres with equal radius, the surface area of each of them is `pi*R^2` . This makes the radius of the sphere `sqrt((pi*R^2)/(4*pi)) = R/2 `

The volume of each sphere is `(4/3)*pi*(R/2)^3` . The total volume of all the four is `(16/3)*pi*R^3/8` = `(2/3)*pi*R^3`

As the volume of the large circle from which the four smaller ones are created is `(4/3)*pi*R^3` there is a decrease in the initial volume to half.

The volume of the smaller spheres is half of the volume of the larger sphere.