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 maximize the volume and use all of the 6-ft board, the bookshelf should have a rectangular shape.
Let's assume the length of the bookshelf is x feet.
To cut the board into five pieces, we will have four shelves of equal width, and the remaining length will be used for the top and bottom of the bookshelf.
Since there are two shelves, the total height of the bookshelf will be 2x.
Given that the board is 24 inches wide (2 feet), and there are four shelves of equal width, each shelf's width will be (24/4) = 6 inches (0.5 feet).
Now, let's calculate the dimensions of the rectangular bookshelf:
1. The total length of the board is 6 feet, and the 2 shelves of width 0.5 feet consume 1 foot (0.5 feet for each shelf). Therefore, the remaining length for the top and bottom of the bookshelf is (6 - 1) = 5 feet.
2. The height of the bookshelf is twice the length of each shelf, which is 2x.
3. The width of the bookshelf is the same as the board's width, which is 0.5 feet.
Now the volume of the bookshelf can be determined using the formula:
Volume = Length x Width x Height
Substituting the values we just calculated, we have:
Volume = 5 feet x 0.5 feet x 2x
Simplifying the expression, we get:
Volume = 5x cubic feet
To maximize the volume, we need to maximize the value of x. Since there are no restrictions mentioned in the problem, we can assume any positive value for x.
Thus, the dimensions of the bookshelf to maximize the volume and use all of the 6-ft board are:
Length: 5 feet
Width: 0.5 feet
Height: 2x feet
Note: The actual value of x can vary depending on the specific requirements of the bookshelf design.
To maximize the volume and use the entire 6-ft board, we need to find the dimensions of the bookshelf that will yield the largest volume. Let's break down the problem step by step.
Step 1: Determine the usable length of the board:
Since the front and back of the bookshelf will not be covered, we can subtract their widths from the total length of the board.
Total usable length = Total length of the board - 2 x Width of the front/back
Total usable length = 6 ft - 2 x 24 in (convert inches to feet by dividing by 12)
Step 2: Determine the dimensions of the bookshelf:
We need to cut the usable length of the board into five equal pieces to create two shelves. Let's represent the length of each piece as "x."
Length of each piece = Total usable length / Number of pieces
Step 3: Calculate the remaining dimension:
Since there will be two shelves, we need to calculate the remaining dimension, which will be the width of the bookshelf.
Width of the bookshelf = Width of the board
Step 4: Calculate the height of the bookshelf:
Since the front and back of the bookshelf will not be covered, the height will be the same as the width of the board.
Height = Width of the board
Now that we have broken down the problem, let's calculate the dimensions of the bookshelf.
Step 1: Convert the inch measurements to feet:
Total usable length = 6 ft - 2 x (24 in / 12)
Total usable length = 6 ft - 4 ft
Total usable length = 2 ft
Step 2: Determine the dimensions of each piece:
Length of each piece = 2 ft / 5
Length of each piece = 0.4 ft
Step 3: Calculate the remaining dimension:
Width of the bookshelf = 24 in / 12
Width of the bookshelf = 2 ft
Step 4: Calculate the height of the bookshelf:
Height = Width of the board = 2 ft
Therefore, the dimensions of the bookshelf to maximize the volume and use the entire 6-ft board are:
Length = 0.4 ft
Width = 2 ft
Height = 2 ft
Apparently we are supposed to ignore the loss due to saw-cuts, a bad idea.
Looks like we have two verticals and 3 horizontal pieces.
let each of the verticals be x and each of the horizontals be y
2x + 3y = 72 inches
y = (72-2x)/3
volume = 24(x)(y)
= 24x(72-2x)/3
= 576x - 16x^2
A parabola opening downwards, so it will have a maximum
Find the vertex and you got it