How can you find the maximal volume of a rectangular box inscribed in a sphere?

Equations (4) and (5) are partial derivatives and have to be equated to zero to maximize the volume of the rectangular box. Hence, the maximal volume of a rectangular box inside the sphere is 8r3/3√3.

How do you find the volume of an open rectangular box?

Understand the volume of a rectangle equals it’s length x width x height. If your box is a rectangular prism or a cube, the only information you need is the box’s length, width, and height. You can then multiply them together to get volume. This formula is often abbreviated as V = l x w x h.

How do you find the maximum value of a sphere?

Formula for the volume of a sphere: V = 4/3*pi*r^3, where r is the radius and V is the volume. So the maximum volume of this sphere is 4,176.2 cubic centimeters.

How much volume of a cube is consumed by an inscribed sphere?

For most practical purposes, the volume inside a sphere inscribed in a cube can be approximated as 52.4\% of the volume of the cube, since V = π6 d3, where d is the diameter of the sphere and also the length of a side of the cube and π6 ≈ 0.5236.

Why is the volume of a box 8xyz?

Clearly the box will have the greatest volume if each of its corners touch the ellipse. Let one of the corners (x, y, z) be in the positive octant, then the box has corners (±x, ±y, ±z) and its volume is V = 8xyz.

What should be the maximum volume of open box?

The formula for volume of the box is V=l×l×h . You can determine the maximum value of this function using graphing calculator. For the maximum, you should get a maximum volume of 13.5 in3 .

How do you find the volume of a box calculator?

To find the volume of a box, simply multiply length, width, and height — and you’re good to go! For example, if a box is 5×7×2 cm, then the volume of a box is 70 cubic centimeters. For dimensions that are relatively small whole numbers, calculating volume by hand is easy.

How do you find the maximum volume of a cylinder inscribed in a sphere?

Let R be the radius of the sphere and let h be the height of the cylinder centered on the center of the sphere. By the Pythagorean theorem, the radius of the cylinder is given by r2=R2−(h2)2. The volume of the cylinder is hence V=πr2h=π(hR2−h34).

How do you find the volume of a sphere inside of a cube?

If s is the side length of the cube, we have that Vcube=s3. Notice that the largest possible sphere that can fit inside the cube is the inscribed sphere, which has radius 12s. Using the volume formula for a sphere, we find that Vsphere=43πr3=43πs38=π6s3.