Maths Mechanics

I need help on this question:

A uniform ladder AB, of mass m and length 2a, has one end 1 on rough horizontal ground. The coefficient of friction between the ladder and the ground is 0.5. The other end B of the ladder rests against a smooth vertical wall. The ladder rests in equilibrium in a vertical plane perpendicular to the wall, and makes an angle of 30° with the wall. A man of mass 5m stands on the ladder which remains in equilibrium. The ladder is modelled as a uniform rod and the man as a particle. The greatest possible distance of the man from A is ka.
Find the value of k.

So far I've drawn it out.

The usual thing is to resolve forces vertically and horizontally and take moments about a point. Without working it through Id expect it to work here, so what are you stuck with??

I've done this working out but idk I feel like everything is wrong

A fair bit

A fair bit

i've done this now idk if its the right way to solve it tho

Getting there, though not sure where the 24 comes from. Note k must be < 2.

Id also get in the habit of resolving horizontally (even if its simple as here) and explicitly saying why limiting friction applies.

ohh yh thank you, not sure why I did 24 it was meant to be 2, I changed that and now i got k as 1.19

Ive not worked it through but you should be able to check by subbing it into the moment equation you start with and verify it balances.

Edit - Im getting k a bit larger so post your working if necessary.

i did https://docs.google.com/drawings/d/e/2PACX-1vRv_XuJA17AKJ6OoWfed74ePgu88CoBvLorwFOYxfnyobRirpGXC3xHXiWrLSm1yp4t8akBV5AJvTRK/pub?w=960&h=720