Abuilding with a rectangular base is to be constructed on a lot in a form of a right triangle with legs 18m and 24m. if the building has one side along the hypotenuse of the triangle, find the dimensions of the base of the building for maximum floor area.

I suggest you to use this picture. It will help you to get what's going on. According to the information you've given, AD = 18m, AG=24m. So we need to know derivates. I suggest you to use 2 common formulas: a*b=A (I'm pretty sure, you know what it is about), and Pythagoras theorem A^2+B^2=C^2. Your main goal is to find out what the lengths of each sides are in relation to changing the length of one side. You've everything to find all the values you need. Hope you'll find it helpful.