Sponge Quality Inspection: Measuring Maximum Pressure with a Rectangular Block

Problem Statement

A quality control department at a sponge factory evaluates the elasticity of sponge products by placing a rectangular metal block of fixed weight on top of a sponge and measuring the depth of the indentation.

A sponge indents more deeply the greater the pressure $P$P applied to its top surface. A rectangular block of weight (gravitational force magnitude) $W$W is placed at rest on a sponge sitting on a horizontal surface. The three edge lengths of the block are $a$a, $b$b, and $c$c, and all faces are smooth.

By changing the orientation of the block (i.e., choosing which face becomes the bottom), the contact area $S$S with the sponge changes, and so does the pressure $P$P on the sponge.

To make the sponge indent as deeply as possible, find the value of the pressure $P$P acting on the sponge. Assume the top surface of the sponge is large enough that the entire bottom face of the block always remains in contact with the sponge.

Constraints

• Weight of block: $W = 39 \, \text{N}$W=39N
• Edge length 1: $a = 0.25 \, \text{m}$a=0.25m
• Edge length 2: $b = 0.20 \, \text{m}$b=0.20m
• Edge length 3: $c = 0.13 \, \text{m}$c=0.13m

Input Format

Find the pressure $P$P in units of $\text{Pa}$Pa and give the numerical value as the answer.