Question

You want to design a set of four congruent pyramids whose combined volume is the same as the volume of a single rectangular pyramid. Why values of l and h for the four square pyramids and what values of l,w, and h for the rectangular pyramid will produce identical volumes

Answer 1

(Square)
Base length-6 units
Height-4 units
Volume-48 cubic units
Volume of 4 square pyramids-192 cubic units
(Rectangular)
Base length-12 units
Base width-8 units
Height-6 units
Volume-192 cubic units

(What I got. Though the answers vary because there's a bunch of ways to do it. This is just one. Hope it helps)

Answer 2

As long as rectangular and square pyramids have the same length and height, the condition will be fulfilled if rectangular pyramid's width is four times square pyramid's length.

Explanation

Reference

l: length

w: width

h: height

Assumption: rectangular and square pyramids have the same length and height.

Volume of a rectangular pyramid

V1 = (1/3)*l*w*h

Volume of a square pyramid

V2 = (1/3)*l^2*h

We need that V2*4 = V1. So

(1/3)*l^2*h*4 = (1/3)*l*w*h

Simplifying

l*4 = w