Max Vol Rect Solid Cut from Sphere: Find Dim & Vol

[paulojomaje]

A rectangular solid of maximum volume is to be cut from a solid sphere of radius r. Determine the dimension of the solid so formed and its volume?
I have defined my function F(l,b,h) as lbh, but i don't know how to define my constraint condition from my question

 

[epenguin]

A rectangular solid of maximum volume is to be cut from a solid sphere of radius r. Determine the dimension of the solid so formed and its volume?
I have defined my function F(l,b,h) as lbh, but i don't know how to define my constraint condition from my question

The constraint - I don't know what stops you seeing this - is that all the corners have to be at the surface of the sphere. Do you know what the equation for a sphere of radius r is (meaning the relation between r and co-ordinates x, y, z holding at all points on the surface)?

Then before you start, think how many independent co-ordinates of the touching points are you going to need? There are 8 points where your solid will touch the sphere surface but you sure need to specify a lot less than that if it is going to be a 'rectangular solid'.

A tip: make equations as simple as possible by choosing simple co-ordinates for points. All points on the sphere are equal as starting points. So you can make your first point (r, 0, 0) for instance.

So now tell us about other points.