I'm looking for hard math, not educated guesses.
Some things I've been looking at:
At what enemy armor level is it more cost effective to build Last Whisper over Black Cleaver?
At what enemy armor level is it more cost effective to build Last Whisper over Blood Thirster?
This is assuming standard AD carry masteries in offense, so +10% armor pen, +6 armor pen. Also assuming 0 armor pen runes (not everyone chooses to run them on ad carries)
I think the other posts pretty much sums it up, it is really situational. I found this which is in an in depth comparison between last whipser and black cleaver.
Copying the tl;dr from the linked article: Everything in League of Legends is situational. Neither Last Whisper nor The Black Cleaver is superior to either in all situations; learn to think analytically and determine in what situations one will do more damage than the other.
Edit: Link contains hard math, result is still "it depends". The lower armor and longer engages the better black cleaver turns out and of course vice versa.
Two things that weren't mentioned is your team composition and the use of abilities in teamfights. Black Cleaver provides 0 armor penetration on the target. Rather it provides an armor reduction debuff on the target that stacks. This means if your team is AD heavy (solo top lane is AD like Riven, Garen, or GP) then having Black Cleaver as a Ranged AD Carry allows you to "tag" the target with the debuff allowing yourself and your AD bruiser to do more damage. Last Whisper on the other hand only affects what you see for the target's armor, their actual armor number is unchanged for everyone else. Also it is effective for your autoattacks as well as your abilities. So for instance if you are trying to poke each other before a full on teamfight starts using abilities like Ashe Volley, Caitlyn's Piltover Peacemaker, etc. Your abilities will have the full effect of a Last Whisper but zero effect of the Black Cleaver debuff, since most abilities do not provide on-hit effects for Ranged AD carries.
Unfortunately you're looking at an incredibly situational and varying part of the game all the while not having given enough information for a "hard math" answer. We don't know either side of the equation (your items or their armour), so until then you provide that, you won't get what you're looking for. I believe however, understanding why people itemize how they itemize is a lot more beneficial than simply relying on a number every single game.
As an AD carry, armour pen past your masteries is only necessary when more than half their team has a significant amount of armour (flat 6 and 10% is quite a bit of armour pen) since you'll be hitting anybody who gets within range.
For example: The enemy team only has armour in the form of Malphite with Randuins and Frozen Heart that charges you. Last Whisper is not going to do a single thing because you're not killing him. However, against a team with 3 bruisers with relatively high armour (~150+ = ~60+% damage reduction), Last Whisper will do wonders.
In short, you should try to stack as much AD and AS as possible until the loosely defined point in the game above, where you absolutely need that 40% ARPen from LW. This is why we see AD carries follow an almost identical build each game (IE/BT -> PD -> LW).
If the enemy is building Thornmail, it's a good idea to get more Armor Pen so you can kill them faster and do less damage to yourself in the end. Whenever your team is very reliant on AD, you should get a Black Cleaver so that all your teammates can do more damage. But, if the enemy has so much armor that you can't hurt them properly, then you should get Armor Pen, otherwise stick with AD.
I know that this topic is old but I was looking for the same answer ;-) Anyway no one is responding to your question so i decided to do the maths.
It started with some Excel graph where i compare 2 runes build (considering only Marks and Quintencences ) and including the 10% armor penetration and 6 flaf from masteries
Full AD build => +16 Flat AD ==> (AD)
Full Pene build => +25 Armor penetration ==> (Pene)
then there is 2 thing that varies : Your total Attack damage (not considering runes) and the armor of the target
Finaly i deduced 2 formula :
1) Calculating a witch point of Attack damage The Armor Penetration build become more effective than the Full AD build for a fix amor value:
Attak = AD((94 + Armor*0.9)/Pene) - 1)
2) Calculating a witch point of Armor The Armor Penetration build become less effective than the Full AD build for a fix Attak damage:
Armor = ((((Attak/AD)+1)*Pene)-94)/0.9
...............................
Example 1 : I'm an AD Carry and there is a tank with 150 Armor
Attak = 16((94 + 150*0.9)/25) - 1) = 130.5
==> So i have to reach 130.5 Damage for my Full Pene build to be more effective.
................................
Example 2 : I'm an AD Carry and i have 150 AD (not considering runes)
Armor = ((((230/16)+1)*25)-94)/0.9 = 183.75
==> So if the target has more than 183.75 Armor my Full AD build is more effective.
.................................
Conclusion : The more Total damage you got the better is the Penetration build. But the more Armor the target has, the less the armor penetration is effective.
