To find the maximum volume of a rectangular box inscribed in a sphere of radius r, we can start by considering the relationship between the box and the sphere.
Let the dimensions of the box be x, y, and z. Since the box is inscribed in the sphere, the corners of the box touch the surface of the sphere. The distance from the center of the sphere to any corner of the box can be expressed using the equation:
√(x² + y² + z²) = r
From here, we can express the volume V of the box as:
V = x * y * z
To maximize this volume, we can use the method of Lagrange multipliers or substitute to eliminate one variable. Let’s use the first method here:
We know from the sphere’s equation that:
z = √(r² – x² – y²)
Substituting z in the volume formula gives us:
V = x * y * √(r² – x² – y²)
This is a function of two variables (x and y). To maximize it, we can take partial derivatives with respect to x and y, set them to zero, and solve for x and y:
After doing the calculations, we find that:
x = y = z = r/√3
The volume can then be expressed as:
V = (r/√3)³ = (r³ / 3√3)
Thus, the maximum volume of the rectangular box inscribed in a sphere of radius r is:
V = (r³ / 3√3)