Answer:
The volume of cube C is
[tex]V=192\sqrt{3}\ units^3[/tex]
Step-by-step explanation:
step 1
Find the diameter of sphere B
we know that
When a cube is inscribed in a sphere, the long diagonal of the cube is a diameter of the sphere
Let
L ----> the length side of cube A
d ----> the diagonal of the base of cube A
D ---> the long diagonal of cube A
Find the diagonal of the base of cube A
Applying the Pythagorean Theorem
[tex]d^2=L^2+L^2[/tex]
we have
[tex]L=4\ units[/tex]
substitute
[tex]d^2=4^2+4^2\\d^2=32\\d=4\sqrt{2}\ units[/tex]
Find the long diagonal of cube A
Applying the Pythagorean Theorem
[tex]D^2=d^2+L^2[/tex]
substitute
[tex]D^2=32+4^2\\D^2=48\\D=4\sqrt{3}\ units[/tex]
step 2
we know that
If sphere B is inscribed in cube C, then the length side of cube C is equal to the diameter of sphere B
Let
c ----> the length side of cube C
we have that
[tex]c=4\sqrt{3}\ units[/tex]
The volume of cube C is equal to
[tex]V=c^3[/tex]
substitute
[tex]V=(4\sqrt{3})^3[/tex]
[tex]V=192\sqrt{3}\ units^3[/tex]
