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?
To find the dimensions of the bookshelf that will maximize its volume while using the entire 6-ft board, we can follow these steps:
Step 1: Convert the 6-ft board into inches. Since 1 ft equals 12 inches, the length of the board is 6 ft × 12 inches/ft = 72 inches.
Step 2: Determine the dimensions of the shelf. Let's assume the shelf's width is x inches. Since there will be two shelves, the total height (thickness) of the bookshelf will be 2x inches.
Step 3: Calculate the length of each shelf piece. Divide the total length of the board (72 inches) minus two times the width of the shelf by 5. This is because there are five pieces in total, and the front and back of the bookshelf will not be covered. So each shelf piece will have a length of (72 inches - 2x inches) / 5.
Step 4: Calculate the volume of the bookshelf. The volume of a rectangular prism is given by V = length × width × height. Since the front and back are not covered, the volume of each shelf piece is x inches × (72 inches - 2x inches) / 5 × 2x inches.
Step 5: Express the volume as a function of x. Multiply the expressions obtained in Step 4 by 2 (since there are two shelves) and simplify to get the volume function in terms of x.
Step 6: Find the critical points of the volume function. Take the derivative of the volume function with respect to x and set it equal to zero. Solve this equation for x to find the critical points.
Step 7: Determine the maximum volume. Substitute the critical points found in Step 6 into the volume function and compare the values to find the maximum volume. The dimensions corresponding to the maximum volume will be the required dimensions of the bookshelf.
Following these steps will ensure that we find the dimensions of the bookshelf that maximize its volume while using the entire 6-ft board.
how wide will the shelves be? If they are 12" wide, he has 12' of board to use. If they are 8" wide, he has 18 ft of board to use.
There will be 2 pieces for the sides, and 4 pieces for top, bottom, and two shelves.