To find the empty space volume inside a cylinder with four spheres just fitting inside, calculate the cylinder's volume and subtract the total volume of the spheres. The cylinder's volume is 1728π cm³ and the four spheres combined volume is 1152π cm³, resulting in an empty space volume of 576π cm³.
To calculate the volume of the empty space inside the cylinder that is not occupied by the spheres, we first have to determine the volume of the cylinder and then subtract the combined volume of the four spheres.
The diameter of the sphere is given as 12 cm, so the radius (r) is half of that, which is 6 cm. With four spheres perfectly fitting inside the cylinder, the height (h) of the cylinder must be four times the diameter of a sphere, which is 48 cm (4 x 12 cm).
First, calculate the cylinder's volume using the formula:
Vcylinder = πr²h
Substitute the known values:
Vcylinder = π x (6 cm)² x 48 cm = 1728π cm³
Next, we need to calculate the volume of a sphere and then find the volume of four such spheres. The formula for the volume of one sphere is:
Vsphere = (4/3)πr³
For a single sphere with a radius of 6 cm:
Vsphere = (4/3)π x (6 cm)³ = 288π cm³
So, the combined volume of four spheres is:
Vfour spheres = 4 x 288π cm³ = 1152π cm³
Finally, subtract the volume of the four spheres from the volume of the cylinder to get the empty space:
Vempty = Vcylinder - Vfour spheres = 1728π cm³ - 1152π cm³ = 576π cm³