Wood Board Length.

Question

I’m trying to find an exact length of a wood board (AB) that is being cut into staves.  

I’ll need to know how to calculate so I can put this into a spreadsheet (Numbers, Excel).

In the example:

1. The board is cut into 16 triangle using 17 angled cuts.
2. The thickness (CD) of the board is known: 0.75”
3. The length of the blue and yellow triangles (staves) is known: 0.5“
4. The angle of the arrows (cuts) is known: 15°
5. The width of the saw blade making the cuts (arrows) is known: 0.125”
6. The grey triangles are waste at each end of the board.

Answer 1

Variables:

1. Number of Staves (n): 16
2. Length of Each Stave (l): 0.5"
3. Thickness of the Board (t): 0.75"
4. Angle of the Cuts (θ): 15°
5. Saw Blade Width (w): 0.125"
6. Grey Triangle Waste at Each End: Account for 2 triangles (one at each end).

Step 1: Understand the Problem

1. Each stave is a triangle, formed by two cuts on each side.
2. The total number of cuts is 17 (including the two end cuts), which produces 16 triangles (staves).
3. The width of the saw blade removes some material with each cut, so this needs to be factored in.
4. There are two waste triangles at both ends of the board.

Step 2: Set up the Calculation

The length of the board is made up of:

1. The total length of the staves (since each stave is 0.5"),
2. The total width lost to the saw blade after making each cut,
3. The grey triangle waste at each end.

Step 3: Calculate Total Stave Length

The total length of all 16 staves combined is:

Total Length of Staves=n×l=16×0.5=8 inches\text{Total Length of Staves} = n \times l = 16 \times 0.5 = 8 \text{ inches}Total Length of Staves=n×l=16×0.5=8 inches

Step 4: Calculate Total Saw Kerf Loss

The saw blade removes material with each cut, so you need to subtract the material lost due to the 17 cuts. For each cut:

Total Saw Kerf Loss=(n+1)×w=17×0.125=2.125 inches\text{Total Saw Kerf Loss} = (n + 1) \times w = 17 \times 0.125 = 2.125 \text{ inches}Total Saw Kerf Loss=(n+1)×w=17×0.125=2.125 inches

Step 5: Account for the Grey Triangle Waste

Assume that the waste at the ends of the board corresponds to the extra two grey triangles:

1. Since no exact size is given for the waste triangles, we'll assume they take up space equivalent to two staves.

Waste Triangle Length=2×l=2×0.5=1 inch\text{Waste Triangle Length} = 2 \times l = 2 \times 0.5 = 1 \text{ inch}Waste Triangle Length=2×l=2×0.5=1 inch

Step 6: Calculate Total Board Length

The total board length ABABAB is the sum of the total length of the staves, the saw kerf loss, and the waste triangles:

Total Board Length=Total Length of Staves+Total Saw Kerf Loss+Waste Triangle Length\text{Total Board Length} = \text{Total Length of Staves} + \text{Total Saw Kerf Loss} + \text{Waste Triangle Length}Total Board Length=Total Length of Staves+Total Saw Kerf Loss+Waste Triangle Length

Substitute the known values:

Total Board Length=8+2.125+1=11.125 inches\text{Total Board Length} = 8 + 2.125 + 1 = 11.125 \text{ inches}Total Board Length=8+2.125+1=11.125 inches

Step 7: Put the Formula in a Spreadsheet

You can implement this in Excel or Numbers using the following formula:

1. Cell for Number of Staves (n): 16
2. Cell for Stave Length (l): 0.5
3. Cell for Saw Blade Width (w): 0.125
4. Cell for Waste Triangle Length (t): 1

In the cell for Total Board Length (AB), input this formula:

=(n×l)+((n+1)×w)+t= (n \times l) + ((n + 1) \times w) + t=(n×l)+((n+1)×w)+t

Which, in your case, simplifies to:

=(16×0.5)+(17×0.125)+1=11.125 inches= (16 \times 0.5) + (17 \times 0.125) + 1 = 11.125 \text{ inches}=(16×0.5)+(17×0.125)+1=11.125 inches

Final Answer

The exact length of the board (AB) is 11.125 inches.