The side length, d, of a cube that contains a sphere depends on the radius, r, of a sphere. Assuming that the faces of the cube are tangent to the sphere, write the volume of the cube as a function of the radius of the sphere.
naturally, the radius of the sphere is half the cube side, or d/2. So, d = 2r.
so, v = d^3 = 8r^3
To find the volume of the cube in terms of the radius of the sphere, we need to establish a relationship between the side length of the cube and the radius of the sphere.
If the cube's faces are tangent to the sphere, it means that the sphere's diameter is equal to the diagonal of the cube. The diagonal of a cube can be found using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In the case of a cube, the diagonal (d) is the hypotenuse, and the sides of the cube (s) are the other two sides. Therefore, we have the equation:
d^2 = s^2 + s^2 + s^2
d^2 = 3s^2
d = √(3s^2)
Now, the diameter of the sphere is equal to the side length of the cube plus twice the radius of the sphere. Mathematically, this can be expressed as:
2r + s = d
s = d - 2r
Substituting the value of d from the previous equation:
s = √(3s^2) - 2r
Now, we can express the volume of the cube as s^3 since all sides of a cube are equal:
Volume = s^3 = (√(3s^2) - 2r)^3
Simplifying this equation, we have:
Volume = (3s^2 - 4√3rs + 12r^2 - 2r√(3s^2))^3
So, the volume of the cube is given by the function:
Volume = (3s^2 - 4√3rs + 12r^2 - 2r√(3s^2))^3