Rate of Cooling of Different Shapes

When an object is heated and left in air, it will gradually cool down as it loses heat to the surrounding environment. The rate at which this cooling occurs depends on several factors, including the shape of the object. In this case, we have a sphere, a cube, and a disc, all made of the same material, quality, and volume, and heated to 900 K. We need to determine which of these shapes will have the lowest rate of cooling.

Surface Area and Cooling

The rate of cooling of an object is directly proportional to its surface area. Objects with larger surface areas will cool down faster than those with smaller surface areas. This is because a larger surface area allows for more heat transfer between the object and its surroundings.

Comparison of Shapes

Cube: A cube has six identical square faces, so its surface area is given by 6 * side^2, where side is the length of the cube's side. The cube has a total surface area of 6a^2, where a is the side length.

Disc: A disc has a circular shape, and its surface area is given by π * radius^2, where π is a constant (approximately 3.14) and radius is the distance from the center of the disc to its outer edge.

Sphere: A sphere has a curved surface, and its surface area is given by 4 * π * radius^2.

Comparison of Surface Areas

To compare the surface areas of the cube, disc, and sphere, we need to consider their volumes. Since the objects have the same volume, we can equate their respective formulas:

6a^2 = π * radius^2 = 4 * π * radius^2

Simplifying this equation, we find:

6a^2 = 4 * π * radius^2

Since the objects have the same volume, their side lengths and radii are related as follows:

a^3 = (4/3) * π * radius^3

Conclusion

From the equation above, we can see that the radius of the sphere is greater than the side length of the cube or the radius of the disc. As a result, the sphere has the largest surface area, followed by the disc, and finally the cube.

Since the rate of cooling is directly proportional to the surface area, the sphere will have the highest rate of cooling, followed by the disc, and finally the cube. Therefore, the correct answer is option 'C' - the sphere will have the lowest rate of cooling.