a solid metal sphere of radius 32cm is melted down and recast to make 64 identical small spheres.what will be the radius of each small sphere?please explain ,thanks
To find the radius of each small sphere, we need to use the concept of conservation of volume.
The volume of the original sphere can be calculated using the formula for the volume of a sphere:
V(original) = (4/3) * π * r(original)^3
where r(original) is the radius of the original sphere.
Since the original sphere is melted down and recast into 64 identical small spheres, the total volume of the small spheres combined should be equal to the volume of the original sphere.
The volume of each small sphere can be calculated using the formula for the volume of a sphere as well:
V(small) = (4/3) * π * r(small)^3
where r(small) is the radius of each small sphere.
Since we have 64 small spheres:
V(total small spheres) = 64 * V(small)
Now, we equate the volume of the original sphere to the total volume of the small spheres:
V(original) = V(total small spheres)
Substituting the formulas for volume into this equation:
(4/3) * π * r(original)^3 = 64 * [(4/3) * π * r(small)^3]
Now, let's solve for the radius of each small sphere.
Dividing both sides of the equation by (4/3) * π:
r(original)^3 = (64 * r(small)^3)
Taking the cube root of both sides:
r(original) = 4 * r(small)
Therefore, the radius of each small sphere is 1/4th (or 25%) of the radius of the original sphere.
Given that the original sphere has a radius of 32 cm, the radius of each small sphere will be:
r(small) = (1/4) * r(original) = (1/4) * 32 cm = 8 cm.
So, the radius of each of the small spheres will be 8 cm.
volume reduce by a factor of 4^3, so
radius reduced by a factor of 4.