If a solid piece of iron in the form of a cuboid of dimensions 49 cm × 33 cm × 24 cm is moulded to form a solid sphere, then radius of the sphere is :

Question

Accepted Answer

To determine the radius of the sphere formed by moulding a solid piece of iron in the form of a cuboid with dimensions 49 cm × 33 cm × 24 cm, we need to follow these steps:

1. **Calculate the Volume of the Cuboid:**
   The volume $ V $ of a cuboid is given by the formula:
   $$
   V = \text{length} \times \text{width} \times \text{height}
   $$
   Substituting the given dimensions:
   $$
   V = 49 \, \text{cm} \times 33 \, \text{cm} \times 24 \, \text{cm} = 38808 \, \text{cm}^3
   $$

2. **Volume of the Sphere:**
   The volume of the sphere must be equal to the volume of the cuboid since the material is conserved. The volume $ V $ of a sphere is given by the formula:
   $$
   V = \frac{4}{3} \pi r^3
   $$
   where $ r $ is the radius of the sphere.

3. **Equate the Volumes and Solve for the Radius:**
   $$
   \frac{4}{3} \pi r^3 = 38808
   $$
   Solving for $ r^3 $:
   $$
   r^3 = \frac{38808 \times 3}{4 \pi}
   $$
   Approximating $\pi$ as 3.14159:
   $$
   r^3 = \frac{116424}{12.56636} \approx 9264
   $$
   Taking the cube root:
   $$
   r \approx \sqrt[3]{9264} \approx 21 \, \text{cm}
   $$

Thus, the radius of the sphere is approximately 21 cm, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2 (23 cm):** If the radius were 23 cm, the volume of the sphere would be:
  $$
  V = \frac{4}{3} \pi (23)^3 \approx 50965.33 \, \text{cm}^3
  $$
  This volume is significantly larger than 38808 cm³, indicating that a radius of 23 cm is incorrect.

- **Option 3 (25 cm):** If the radius were 25 cm, the volume of the sphere would be:
  $$
  V = \frac{4}{3} \pi (25)^3 \approx 65449.85 \, \text{cm}^3
  $$
  This volume is much larger than 38808 cm³, making a radius of 25 cm incorrect.

- **Option 4 (19 cm):** If the radius were 19 cm, the volume of the sphere would be:
  $$
  V = \frac{4}{3} \pi (19)^3 \approx 28753.33 \, \text{cm}^3
  $$
  This volume is smaller than 38808 cm³, indicating that a radius of 19 cm is incorrect.

Therefore, based on the calculations and logical reasoning, Option 1 (21 cm) is indeed the correct answer.