This is a bit like all those ladder-to-wall-across-a-fence problems, but you're minimising something other than the ladder.
Start by relabelling the nice picture at http://www.mathhelpforum.com/math-he...dder-wall.html. Let's use capitals for constants.
Z = length of the ladder / glass roof
H = height of the fence / tallest person.
x = distance of roof-end from wall
y = height of other roof-end up wall, but
and finally, u = usable floor.
Use similar triangles to see that the ratio of (x - u) to H is the same as that of x to y.
Express u as a function of x, with constants Z and H. Differentiate u with respect to x and set to zero to find the best x.
Just in case a picture helps differentiate...
... is the chain rule, here wrapped inside...
... the product rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).
In the bottom row, the fourth balloon along (second from left in the big network) is unchanged on its way down from the top row (it wasn't its turn - by the product rule - to get differentiated) but I've tweaked it for the sake of a common denominator.
Then set the derivative to zero. You can thus offer your parents an equation involving x = the optimum extension from the wall, and H and Z for which (presumably) they know the values. I doubt whether we can solve algebraically for x in terms of H and Z... but I'm not at all sure.
Don't integrate - balloontegrate!
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