What's the largest volume/dimensions of given triangular prism?

Question

What are the dimensions of the biggest possible volume (absolute maximum) of a triangular prism given the length and girth must equal 108 inches?

Thanks! I struggle with optimization to such a high degree.

Answer 1

Volume of a triangle prism is

 

V = (1/4)h[-a⁴ + 2(ab)² + 2(ac)² - b⁴ + 2(bc)² - c²]^1/2

 

where:

h = height of prism

a, b, and c are sides of triangle section

 

Girth means longest side of base triangle section.  In other words it is the altitude of the base triangle section.

 

What you need to from here is get the volume formula to have only one variable.  Then you can derive V and set that derivative equal to zero to solve for that variable.

 

a = b = c = girth = 108

 

Based on the diagram you were given, come up with a formula to represent the girth (altitude) using the variables a, b, and, c.