What is the largest cube I could make out of a 4’x8’ sheet of 3/4” plywood, accounting for overlap of the edges?
What is the largest cube I could make out of a 4’x8’ sheet of 3/4” plywood, accounting for overlap of the edges?
user
Based on the dimensions you provided, the largest cube you could make out of a 4'x8' sheet of 3/4" plywood, accounting for the overlap of the edges, would be 4 cubic feet. In order to calculate this, we can assume that you fold the sheet into six even strips of width 4' (24") and then stack them on top of each other. Additionally, it is important to remember that the overlap of the edges would further decrease the amount of material available for the cube.
assistant
To determine the maximum size of a cube that can be made from a 4' x 8' sheet of 3/4" plywood, accounting for edge overlap, we need to consider a few factors:

1. The thickness of the plywood: A 3/4" thick plywood will add an additional 3/4" to the overall height of the cube.

2. The minimum edge overlap: To ensure a strong and stable cube, we'll assume a minimum edge overlap of 1/2" (1/4" on each side).

3. The shape of the cube: A cube has six identical faces, so we'll calculate the maximum dimension for one face and then multiply it by 3 to get the total volume.

Now, let's work through the calculation:

1. Calculate the available area of the plywood sheet:

   A 4' x 8' sheet of plywood has an area of 32 square feet (4 feet x 8 feet = 32 sq. Ft.).

2. Subtract the thickness of the plywood from the available area:

   Since the plywood is 3/4" thick, we need to subtract 0.75 inches (3/4 inch) from the length and width of the sheet:

   Available area - Thickness = (4 feet - 0.75 inches) x (8 feet - 0.75 inches) = 31.25 square feet

3. Divide the available area by the area of one face (a square) to find the maximum dimension for one face:

   Area of one face = (Maximum dimension / 2)²

   Solving for the maximum dimension:

   Maximum dimension = sqrt(2 x Area / 1) = sqrt(2 x 31.25 sq. Ft. / 1 sq. Ft.) ≈ 5.6 feet

4. Multiply the maximum dimension by 3 to get the total volume:

   Total volume = (5.6 feet)³ = 175.36 cubic feet

5. Account for the edge overlap:

   To account for the 1/2" edge overlap, we need to subtract twice the overlap from the maximum dimension:

   Maximum dimension with overlap = 5.6 feet - (2 x 0.5 inches / 12 inches per foot) = 5.5 feet

6. Calculate the adjusted total volume:

   Adjusted total volume = (5.5 feet)³ = 168.25 cubic feet

Therefore, the largest cube you could make out of a 4' x 8' sheet of 3/4" plywood, accounting for a minimum edge overlap of 1/2", would have a maximum side length of approximately 5.5 feet and a total volume of 168.25 cubic feet.
What is the largest cube I could make out of a 4’x8’ sheet of 3/4” plywood, accounting for overlap of the edges?
user
To determine the maximum size of a cube that can be made from a 4' x 8' sheet of 3/4" plywood, accounting for edge overlap, we need to consider a few factors:

1. The thickness of the plywood: A 3/4" thick plywood will add an additional 3/4" to the overall height of the cube.

2. The minimum edge overlap: To ensure a strong and stable cube, we'll assume a minimum edge overlap of 1/2" (1/4" on each side).

3. The shape of the cube: A cube has six identical faces, so we'll calculate the maximum dimension for one face and then multiply it by 3 to get the total volume.

Now, let's work through the calculation:

1. Calculate the available area of the plywood sheet:

   A 4' x 8' sheet of plywood has an area of 32 square feet (4 feet x 8 feet = 32 sq. Ft.).

2. Subtract the thickness of the plywood from the available area:

   Since the plywood is 3/4" thick, we need to subtract 0.75 inches (3/4 inch) from the length and width of the sheet:

   Available area - Thickness = (4 feet - 0.75 inches) x (8 feet - 0.75 inches) = 31.25 square feet

3. Divide the available area by the area of one face (a square) to find the maximum dimension for one face:

   Area of one face = (Maximum dimension / 2)²

   Solving for the maximum dimension:

   Maximum dimension = sqrt(2 x Area / 1) = sqrt(2 x 31.25 sq. Ft. / 1 sq. Ft.) ≈ 5.6 feet

4. Multiply the maximum dimension by 3 to get the total volume:

   Total volume = (5.6 feet)³ = 175.36 cubic feet

5. Account for the edge overlap:

   To account for the 1/2" edge overlap, we need to subtract twice the overlap from the maximum dimension:

   Maximum dimension with overlap = 5.6 feet - (2 x 0.5 inches / 12 inches per foot) = 5.5 feet

6. Calculate the adjusted total volume:

   Adjusted total volume = (5.5 feet)³ = 168.25 cubic feet

Therefore, the largest cube you could make out of a 4' x 8' sheet of 3/4" plywood, accounting for a minimum edge overlap of 1/2", would have a maximum side length of approximately 5.5 feet and a total volume of 168.25 cubic feet.
assistant
What is the largest cube I could make out of a 4’x8’ sheet of 3/4” plywood, accounting for overlap of the edges?
user
To maximize the volume of a cube that can be made from a 4'x8' sheet of 3/4" plywood, accounting for the overlap of the edges, we need to consider the way the sheet is oriented and how much material is wasted due to the overlap.

Let's begin by orienting the 4'x8' sheet so that the longer dimension is along the 4' side. This allows us to maximize the length and width of the resulting cube.

First, we need to account for the thickness of the plywood. Since the plywood is 3/4" thick, we will lose 3/4" on each edge when cutting the cube. Therefore, the actual dimensions of the sheet to work with are (4'-3/4")x(8'-3/4").

To find the largest possible edge length of the cube, we can use the formula for the volume of a cube, which is V=L^3, where L is the edge length. We can rearrange this to solve for L:

L = sqrt(V/(4*pi))

Assuming we want to maximize the volume, we is approximately 3.14159. Let's plug in the dimensions we have:

L = sqrt(V/(4*pi))
V = L^3 \* 4 \* pi

Since the actual dimensions of the sheet are (4'-3/4")x(8'-3/4"), we have an available volume of (4'-3/4")^3 \* 4 \* pi - (8'-3/4")^
assistant