Worked Solution: Volume of a Sphere with Radius 5
Welcome to the complete, step-by-step mathematical walkthrough for finding the volume, surface area, and fluid capacity of a sphere having a radius of exactly r = 5 units.
⚡ Quick Numerical Summary:
- Radius (r): 5 units
- Diameter (d): 10 units
- Exact Volume (with π): 166.67π cubic units
- Decimal Volume: ≈ 523.60 cubic units
- Exterior Surface Area: ≈ 314.16 square units
Step-by-Step Mathematical Solution
| Step | Action | Formula & Algebraic Working | Result |
|---|---|---|---|
| Step 1 | State the standard formula | V = ⁴⁄₃ × π × r³ | Formula stated |
| Step 2 | Substitute radius (r = 5) | r³ = (5)³ = 5 × 5 × 5 | 125 |
| Step 3 | Multiply by constant ⁴⁄₃ | ⁴⁄₃ × 125 = 166.67 | 166.67π (Exact) |
| Step 4 | Evaluate with π ≈ 3.14159265 | 166.67π × 3.14159265 | 523.60 |
Real-World Applications of r = 5
A sphere of radius 5 cm has a diameter of 10 cm. In everyday physics and industry, spheres of this size approximate common sports equipment, ball bearings, optical lenses, or storage vessels. Its capacity in liquid measure equals 0.524 Liters.