SOLUTION: James has a 6-ft board that is 24 in. wide. He wants to cut it into five pieces to make a small bookshelf with two shelves. The front and back will not be covered. What should be t

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Question 1109690: James has a 6-ft board that is 24 in. wide. He wants to cut it into five pieces to make a small bookshelf with two shelves. The front and back will not be covered. What should be the dimensions of the bookshelf to maximize the volume and use all of the 6-ft board?
Answer by greenestamps(13215) About Me  (Show Source):
You can put this solution on YOUR website!


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.

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%286-3x%29%2F2+=+%286x-3x%5E2%29%2F2+=+3x-%283%2F2%29x%5E2

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.