A spherical gemstone just fits inside a plastic cube with edges 12cm. a) calculate the volume of gemstone, to the nearest cubic centimeter. b) how much empty space is there.

volume of gemstone = 905 cm^3

volume of empty space = 823 cm^3

Step-by-step explanation:

volume of cube = s^3, where s = length of edge

volume of sphere = (4/3) (pi) r^3, where r = radius of sphere

The cube has a 12-cm edge. The sphere fits tightly inside the cube, so the diameter, d, of the sphere is 12 cm. The radius is half the diameter, so radius = r = diameter/2 = 12 cm/2 = 6 cm.

a)

volume of sphere = (4/3) (pi) r^3

volume of sphere = (4/3) (3.14159) (6 cm) ^3

volume of sphere = 905 cm^3

b)

The empty space is the difference between the volume of the cube and the volume of the sphere.

volume of cube = s^3

volume of cube = (12 cm) ^3

volume of cube = 1728 cm^3

empty space = volume of cube - volume of sphere

empty space = 1728 cm^3 - 905 cm^3

empty space = 823 cm^3

a) (V) = 904.78 of a sphere = 288pi diameter = 12

(V) = 1728cm^3 of a cube = face diagonal = 16.9cm

b) Difference Volume = 1728-904.78 = 823.22cm^3

Step-by-step explanation:

To find volume of an inscribed sphere within a square cube

We use 4 π/3 * r^3 for the equation

As Radius = 6 = 6cm this is the only thing plugged into the equation to create a division first then a multiplication square of radius and then a multiplication. 4pi / 3 * 6^3

r^3 = 216

4pi/3 = 4.18

4.18 * 216 = 904.78

This means the answer is 288 pi cm^3.