A solid metallic sphere of diameter 28 cm is melted and recast into a number of smaller cones, each of diameter 4 2/3 cm and height 3 cm. Find the number of cones so formed.

To solve this problem, we will use the principle of conservation of volume. The volume of the metallic sphere will be equal to the total volume of the smaller cones formed when the sphere is melted and recast.

Step 1: Calculate the volume of the metallic sphere
The formula for the volume of a sphere is:

V_sphere = (4/3) * π * r^3

Where:

r is the radius of the sphere
Given that the diameter of the sphere is 28 cm, the radius (r) will be:

r = 28 / 2 = 14 cm

Now, calculate the volume of the sphere:

V_sphere = (4/3) * π * (14)^3 V_sphere = (4/3) * π * 2744 V_sphere = 3658.67π cubic cm

Step 2: Calculate the volume of one cone
The formula for the volume of a cone is:

V_cone = (1/3) * π * r^2 * h

Where:

r is the radius of the cone's base
h is the height of the cone
The diameter of each cone is given as 4 2/3 cm. First, convert this to an improper fraction:

4 2/3 = 14/3 cm

The radius (r) of the cone will be half of the diameter:

r = (14/3) / 2 = 7/3 cm

The height (h) of each cone is given as 3 cm. Now, calculate the volume of one cone:

V_cone = (1/3) * π * (7/3)^2 * 3 V_cone = (1/3) * π * (49/9) * 3 V_cone = (1/3) * π * 49/3 V_cone = 49π/9 cubic cm

Step 3: Find the number of cones
To find the number of cones, divide the volume of the sphere by the volume of one cone:

Number of cones = V_sphere / V_cone Number of cones = (3658.67π) / (49π/9) Number of cones = (3658.67 * 9) / 49 Number of cones = 32928.03 / 49 Number of cones ≈ 672

Therefore, the number of cones that can be formed is approximately 672.