The context for this problem is identical to my first question. Unfortunately my previous result for $L$ was inexact since it was a relatively simple approximation. This time I have managed to produce a very convincing approximation for $L$. Alike last time, solving for $W$ provides the $n$th solution if you find the $n$th root of a specific [octive] polynomial:
$96x^8-192Lx^7+(768π^2 A^2+128L^2 ) x^6+(-64L^3-640A^2 π^2 L) x^5+(2655π^4 A^4-640L^2 π^2 A^2+32L^4 ) x^4+(384A^2 L^3 π^2-578A^4 Lπ^4 ) x^3+(4476π^6 A^6-2760L^2 π^4 A^4+128L^4 π^2 A^2 ) x^2+(456A^6 Lπ^6-278A^4 L^3 π^4 )x+(3136π^8 A^8-3700L^2 π^6 A^6+1089L^4 π^4 A^4 )=0$
Since my investigation is based on the assumption that there is only one value for $W$, given the values of $A$ and $L$, ergo there should only be two real roots (one should be the positive value for $W$ and the other should be the negative value for $W$.)
My question is, can you find the positive real root for the eight order polynomial above?