The lateral surface area of a cube is 420 cm$^2$ less than the lateral surface area of a cylinder. The height of the cylinder is $2x$ cm, and the radius of the cylinder is $x$ cm. The side of the cube is equal to the radius of the cylinder. Find the approximate area of a circle with radius $x + 3$ cm.

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**Solution:**

1. Let the side of the cube be $x$ cm (given equal to the radius of the cylinder).

2. Lateral surface area (LSA) of the cube is given by:
   $$\text{LSA}_{cube} = 4x^2$$

3. LSA of the cylinder is:
   $$\text{LSA}_{cylinder} = 2\pi r h = 2\pi x (2x) = 4\pi x^2$$

4. According to the problem, the cube's lateral surface area is 420 cm$^2$ less than the cylinder's lateral surface area:
   $$4\pi x^2 - 4x^2 = 420$$

5. Factor out $4x^2$:
   $$4x^2(\pi - 1) = 420$$

6. Solve for $x^2$:
   $$x^2 = \frac{420}{4(\pi - 1)} = \frac{420}{4 \times (3.1416 - 1)} = \frac{420}{4 \times 2.1416} = \frac{420}{8.5664} \approx 49.03$$

7. Calculate $x$:
   $$x = \sqrt{49.03} \approx 7 \text{ cm}$$

8. The radius of the required circle is $x + 3 = 7 + 3 = 10$ cm.

9. Area of the circle is:
   $$\pi r^2 = \pi \times 10^2 = 100\pi \approx 100 \times 3.1416 = 314.16 \text{ cm}^2$$

10. However, the options do not have 314.16 as the correct answer. Let's re-check step 4:

The problem states the cube's lateral surface area is 420 cm$^2$ less than the cylinder's lateral surface area, so:
   $$\text{LSA}_{cylinder} - \text{LSA}_{cube} = 420$$

So,
   $$4\pi x^2 - 4x^2 = 420$$

This matches our earlier equation.

11. Recalculate $x^2$:
   $$x^2 = \frac{420}{4(\pi - 1)} = \frac{420}{4 \times 2.1416} = \frac{420}{8.5664} \approx 49.03$$

12. So $x \approx 7$ cm.

13. Area of circle with radius $x + 3 = 10$ cm is approximately 314.16 cm$^2$, which corresponds to option (d).

14. But option (d) is 314 cm$^2$, which is close to our calculation.

**Answer:** 314 cm$^2$ (Option d)

**Note:** The correct option based on calculation is (d) 314 cm$^2$. However, the initially marked correct option was (b) 396 cm$^2$. This seems to be a mistake. The correct answer is option (d).

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