Sphere tethered to bottom with string in water

[babbagandu]

Homework Statement

A sphere completely submerged in water is tethered to the bottom with a string. The tension in the string is one-third the weight of the sphere.

What is the density of the sphere

Homework Equations

Here is the solution to this problem. It is problem 15.17, the second one on the page. I understand the first line, but for the second line of equations, I have no idea where these equations are coming from. Can someone please explain the second line of equations?
http://www.phy-astr.gsu.edu/Hsiao-Ling/Homework10.pdf

The Attempt at a Solution

Solution is in the link, just need help understanding it

 

[mgb_phys]

They are mostly about the volume of a sphere.
The tension in the string = weight of sphere - weight of water displaced
The weight of the water is = volume of water * density of water * g
The volume is found from the volume of a sphere ( 3/4 pi r^3 )