Question 1109690
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Since the board and therefore the bookshelf is of uniform width, we can work the problem in 2 dimensions instead of 3, maximizing the area of the front of the bookshelf.<br>
If x is the length of the two shelves and the top, then the length of each side of the bookcase is (6-3x)/2.  Then the area of the front of the bookcase is
{{{x(6-3x)/2 = (6x-3x^2)/2 = 3x-(3/2)x^2}}}<br>
You can find the value of x that maximizes the area using either calculus or analytic geometry; then you can determine the dimensions of the bookcase.