Reverse Engineering: Solving for Radius
In many practical and standardized testing scenarios, you are given the volume of a sphere and tasked with finding its radius. This requires solving the volume equation for r algebraically.
Algebraic Derivation
- Start with:
V = (4/3)πr³ - Multiply both sides by 3:
3V = 4πr³ - Divide both sides by 4π:
r³ = (3V) / (4π) - Take the cube root of both sides: r = ³√( (3V) / (4π) )
Worked Reverse Problem
A water tank holds V = 500 m³ of water in a spherical vessel. Find its internal radius.
1. Compute 3V: 3 × 500 = 1500
2. Compute 4π: 4 × 3.14159 = 12.56637
3. Divide: 1500 / 12.56637 = 119.366
4. Cube root: ³√(119.366) ≈ 4.92 meters