What is the volume of the largest sphere that can be carved out of a wooden cube of sides 21 cm? ($\pi = \frac{22}{7}$)

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**Solution:**

1. The largest sphere that can be carved out of a cube will have its diameter equal to the side length of the cube.

2. Given the cube side length $ s = 21 \text{ cm} $, the diameter of the sphere $ d = 21 \text{ cm} $.

3. The radius of the sphere is half the diameter:
   $$
   r = \frac{d}{2} = \frac{21}{2} = 10.5 \text{ cm}
   $$

4. The volume of a sphere is given by:
   $$
   V = \frac{4}{3} \pi r^3
   $$

5. Substitute $ \pi = \frac{22}{7} $ and $ r = 10.5 $:
   $$
   V = \frac{4}{3} \times \frac{22}{7} \times (10.5)^3
   $$

6. Calculate $ (10.5)^3 = 10.5 \times 10.5 \times 10.5 = 1157.625 $.

7. Now,
   $$
   V = \frac{4}{3} \times \frac{22}{7} \times 1157.625 = \frac{88}{21} \times 1157.625
   $$

8. Calculate:
   $$
   V = \frac{88}{21} \times 1157.625 = 4851 \text{ cm}^3
   $$

**Answer:** The volume of the largest sphere is $ 4851 \text{ cm}^3 $.

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