A solid cube and a solid sphere of the same material have equal surface area. Both are at the same temperature `120^(@)C`, then

Options

A. Both the cube and the sphere cool down at the same rate
B. The cube cools down faster than sphere
C. The sphere cools down faster than the cube
D. Whichever is having more mass will cool down faster

Answer

To solve the problem, we need to analyze the cooling rates of a solid cube and a solid sphere made of the same material, given that they have equal surface areas and are at the same temperature.

### Step-by-Step Solution:

1. **Understanding Surface Area**:
- The surface area of a cube (A_cube) is given by \( A_{cube} = 6a^2 \), where \( a \) is the length of a side of the cube.
- The surface area of a sphere (A_sphere) is given by \( A_{sphere} = 4\pi r^2 \), where \( r \) is the radius of the sphere.
- Since it is given that the surface areas are equal, we can set them equal to each other:
\[ 6a^2 = 4\pi r^2 \]

2. **Finding the Ratio of Dimensions**:
- Rearranging the equation gives:
\[ \frac{a^2}{r^2} = \frac{2\pi}{3} \]
- Taking the square root of both sides gives:
\[ \frac{a}{r} = \sqrt{\frac{2\pi}{3}} \approx 1.1447 \]
- This indicates that the side length of the cube is greater than the radius of the sphere.

3. **Calculating Volume**:
- The volume of the cube (V_cube) is given by \( V_{cube} = a^3 \).
- The volume of the sphere (V_sphere) is given by \( V_{sphere} = \frac{4}{3}\pi r^3 \).
- To find the ratio of their volumes, we can express both volumes in terms of \( r \):
\[ V_{cube} = a^3 = \left( \sqrt{\frac{2\pi}{3}} r \right)^3 = r^3 \left( \frac{2\pi}{3} \right)^{3/2} \]
- Therefore, the volume ratio becomes:
\[ \frac{V_{sphere}}{V_{cube}} = \frac{\frac{4}{3}\pi r^3}{\left( \frac{2\pi}{3} \right)^{3/2} r^3} = \frac{4\pi}{\left( \frac{2\pi}{3} \right)^{3/2}} \]

4. **Simplifying the Volume Ratio**:
- Simplifying this expression gives:
\[ \frac{V_{sphere}}{V_{cube}} = \frac{4\pi}{\frac{2\sqrt{2}\pi^{3/2}}{3\sqrt{3}}} = \frac{4 \cdot 3\sqrt{3}}{2\sqrt{2}\pi^{1/2}} = \frac{6\sqrt{3}}{\sqrt{2}\pi^{1/2}} \]
- This ratio is greater than 1, indicating that the sphere has a larger volume than the cube.

5. **Cooling Rate Analysis**:
- According to Newton's law of cooling, the rate of heat loss is proportional to the surface area and the temperature difference between the object and its surroundings.
- Since both objects have the same surface area and are at the same initial temperature, the rate of heat emission will be the same.
- However, the mass of the objects will differ due to their volume difference. The sphere, having a larger volume, will have a larger mass.

6. **Conclusion**:
- Since the sphere has more mass, it will take longer to cool down compared to the cube. Therefore, the solid cube will cool down faster than the solid sphere.

### Final Answer:
The solid cube will cool down faster than the solid sphere.