I was preparing for my maths test . And preparing application of derivative (theory based question ) there I saw a problem of proving rectangle of maximum area in a circle is square . So there were two variables and I was able to find one in terms of other by applying Pythagoras since diagonal of rectangle is diameter of circle.
But soon I thought of a problem that similarly a cuboid of maximum volume in a sphere would be a cube .
But when I went of proving there were three different variables for length ,breadth and height and i m not able to convert two of them in one variable. So it is becoming a constant problem for me . So please guide how should I move further so that I can change three variables into one .
I hope you guys understood my problem .
Proving that cuboid of maximum volume in a sphere is a cube.
applicationsderivativesmaxima-minimaspheresvolume
Best Answer
Make the rectangular prism's length, depth and height a function of x. This is doable because the prism fits into a sphere. Consider a view looking onto one face of the prism with the origin of a coordinate grid at the center of the circumscribing sphere. The length is $2x$, and the height and depth are both $2\sqrt{r^2 - x^2}$, where $r$ is the radius of the sphere.