To find the diameter of the metallic sphere and its surface area, we need to consider the volume and surface area of the cones.

Step 1: Find the total volume of the cones
The volume of a cone is given by the formula V = (1/3) * π * r^2 * h, where r is the radius of the base and h is the height.
In this case, the diameter of the base is given as 3.5 cm. Therefore, the radius (r) is half the diameter, which is 1.75 cm. The height (h) is given as 3 cm.
Using the formula, the volume of one cone is V = (1/3) * π * (1.75 cm)^2 * 3 cm.
Multiplying the volume of one cone by the total number of cones, we get the total volume of the cones: Total Volume = 504 * V.

Step 2: Find the diameter of the sphere
The volume of a sphere is given by the formula V = (4/3) * π * r^3, where r is the radius of the sphere.
Since the volume of the cones and the sphere will be the same (due to the conservation of volume), we can equate the two volumes:
504 * V = (4/3) * π * r^3.
Simplifying the equation, we get r^3 = (3/4) * (504 * V) / π.
Taking the cube root of both sides, we can find the radius (r) of the sphere.
Once we have the radius, we can calculate the diameter of the sphere by multiplying the radius by 2.

Step 3: Find the surface area of the sphere
The surface area of a sphere is given by the formula A = 4 * π * r^2, where r is the radius of the sphere.
Using the radius found in the previous step, we can calculate the surface area of the sphere by substituting the value of r in the formula.

In conclusion, by following the above steps, you can find the diameter of the metallic sphere and its surface area using the given information about the cones.