First of all let us note that the foreman said "engineer". That means that he doesn't expect a mathematical solution that in this case would lead to an equation of degree 4!  Thank you very much Mr. foreman!
So what we are out for is an approximate solution that is good to 3 decimals! Now to some preliminary definitions. They all are given in the figure below.

and by using the Pythagorean theorem that

Now to the engineering part. The solution we are looking for is close to 9 as it would be if the ladder was upright at the wall. Thus 1/x is small as compared to 1 and therefore we can assign a recursive formula that uses the "old" value of x in the "1/x" term when we compute the "new" value of x. The formula is given below:

Using this formula and starting with the approximate solution x=9 the process quickly converges as in the table below.
 
 

Thus after 2 iterations the solution is stable and with an engineering satisfaction to 3 decimals equal to 9.938 units.
 
 

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