Answer:
a) The length of its edge is 8 units.
b) The volume of the sphere is approximately 268.083 cubic units.
c) The fraction of the cube that is taken up by the sphere is 0.524, whose equivalent in percentage form is 52.4 %.
Step-by-step explanation:
a) The volume of a cube ([tex]V_{c}[/tex]), measured in cubic units, is defined by the following expression:
[tex]V_{c} = L^{3}[/tex] (1)
Where [tex]L[/tex] is the side length of the cube, measured in units.
If we know that [tex]V_{c} = 512\,u^{3}[/tex], then the length of its edge is:
[tex]L = \sqrt[3]{V_{c}}[/tex]
[tex]L = 8\,u[/tex]
The length of its edge is 8 units.
b) The volume of a sphere ([tex]V_{s}[/tex]), measured in cubic units, that fits snugly inside this cube is defined by:
[tex]V_{s} = \frac{\pi}{6}\cdot L^{3}[/tex] (2)
If we know that [tex]L = 8\,u[/tex], then the volume of the sphere is:
[tex]V_{s} \approx 268.083\,u^{3}[/tex]
The volume of the sphere is approximately 268.083 cubic units.
c) The fraction of the cube taken by the sphere is the ratio of the volume of the sphere to the volume of the cube. That is to say:
[tex]r = \frac{V_{s}}{V_{c}}[/tex]
If we know that [tex]V_{c} = 512\,u^{3}[/tex] and [tex]V_{s} \approx 268.083\,u^{3}[/tex], then the fraction of the cube that is taken up by the sphere is:
[tex]r = 0.524[/tex]
The result in percentage form is calculated by multiplying the result above by 100.
[tex]\% r = 52.4\,\%[/tex]
The fraction of the cube that is taken up by the sphere is 0.524, whose equivalent in percentage form is 52.4 %.
