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I really can't follow what you're saying. I just want to know where that expression for the height comes from. So I called it h to get the total area of the triangle as h(ax+b)/2. Total total yield of this area will be hm(ax + b)/2.
So I can appreciate that ax must be something relating that smaller triangle to the height, and if I set h = x, I get m(ax^2 + bx)/2 for the total yield. Substituting h = 2x/m gives ax^2 + bx which is the area of two quadrilaterals with the same height of x and 2 sides of ax and b. So the yield, which should be a product of area and the coefficient m is now rendered as the areas of two squares without having anything to do with that coefficient anymore. Taking the height to be x again and the bases as they are, the total yield for the aggregate quadrilateral is m(ax^2 + bx), and for the triangles would be that over 2. Rearranging gives ax^2 + bx = 2yield /m. So if the yield of the quadrilateral (divided by m) of height x is ax^2 + bx, then the height at which the yeild of the triangles is equal to that is 2x/m. I can see all that but I just can't grasp what on earth is going on and its doing my head in