I was working on Lagrange Multipliers but I want to find another method using Geometry..
From calculus book the question is, with the information given,
"Find the dimensions of the rectangular box of maximum volume that can be inscribed inside the sphere $x^2 +y^2 +z^2 = 4$"
I understand how to do it their way and I do not seek an optimization approach or a Lagrange Multiplier approach but a geometric one with shapes/triangles and algebra and reason behind the process.
The answer is that the volume consists of all like edges, $\frac{4}{\sqrt{3}}$ and I can work backwards geometrically from this answer given in the book but I want to know if you can work forward from the information given only by the initial problem using geometry.
Best Answer
Transpose the problem to two dimensions to better see what's going on. What is the largest square we can fit inside a circle of radius two?
Question: What is the longest line inside a circle of radius two?
Notice that the largest rectangle that can be put inside a circle is a square whose diagonal is a diameter of the circle. Now we move this reasoning up to three dimensions.
In three dimensions, we know we want some sort of cube of side length $x$ with volume $x^3$. Again, the diagonal of the cube must coincide with the longest distance in the sphere, namely the diameter of $4$. Now apply the pythagorean theorem twice to the cube to find the diagonal of the cube: $$x^2 + x^2 + x^2 = (4)^2 \implies x = \frac{4}{\sqrt{3}} \implies x^3 = \frac{64}{3\sqrt{3}}$$ So we are done. The key thing to remember from this is that working one dimension lower often gives you a clear path to follow in the higher dimension. In this case, we see that we can take a cross section of the sphere in question and just work with a circle, as the logic transfers directly.