A sphere of surface area 16𝜋
square centimeters is inscribed in a cube. Find the volume of the cube.
The key to this question is being
able to relate the dimensions of the cube and the sphere. We note that the sphere is
inscribed in the cube, which means that the sphere is touching each face of the cube
without any gaps. Therefore, the diameter of the
sphere must equal the length of the cube. In other words, the length of the
cube is two times the radius. This is shown in the figure.
We have been given the surface area
of the sphere, 16𝜋 square centimeters. Recall that if the radius of a
sphere is 𝑟, then its surface area, 𝑆, is given by four 𝜋𝑟 squared. Substituting 𝑆 equals 16𝜋, we
have 16𝜋 equal to four 𝜋𝑟 squared. Now we will solve for 𝑟 by
dividing through by four 𝜋. This gives us four equal to 𝑟
squared. Then, we can obtain 𝑟 by taking
the square root of four. So, 𝑟 equals two centimeters.
We are looking for the volume of
the cube, which is given by the formula 𝑉 equals 𝑙 cubed. As mentioned earlier, the length of
the cube is two times the radius of the inscribed sphere, so we have 𝑙 equal to
four centimeters. Now that we have the length of the
cube, we can obtain its volume by substituting 𝑙 equals four. So, the volume of the cube
described is 64 cubic centimeters.