Calculate Board Feet from Log Diameter and Length | Basic Unit Conversions

[youngstudent16]

Homework Statement

Calculate the approximate number of board feet (one board foot is defined as a volume of $1 in \times 1 ft \times1 ft$ of lumber that would be available in a log that has a diameter of $34 cm$ and a length of $3 m$. Assume the log is a right cylinder and that the saw mill must make the log into a rectangular prism before cutting it into boards.

Homework Equations

The volume of a cylinder

The Attempt at a Solution

I converted the diameter of meters and I know I can use the diameter of the log to figure out the length of the sides of the square that will eventually make up.

So, the diagonal of the square is the diameter of the circle. The circle has diameter $34 cm$, so the square has side length $34 \sin{45} =17\sqrt{2} cm$

Do I now use trig to find the last missing side knowing that side length plus the $3m$ length given and the diagonal known. Multiply those together and convert to the right units before I do so?

 

[William White]

What is the biggest quadrilateral you can fit in a circle of diameter 34cm that has a side of 1ft?

http://www.mathopenref.com/quadrilateralinscribedarea.html

 

[billy_joule]

What is the biggest quadrilateral you can fit in a circle of diameter 34cm that has a side of 1ft?

http://www.mathopenref.com/quadrilateralinscribedarea.html

I'n not so sure that the question implies the rectangle needs a dimension of 1ft. the board foot is just a volume, the shape can vary.

I converted the diameter of meters and I know I can use the diameter of the log to figure out the length of the sides of the square that will eventually make up.

So, the diagonal of the square is the diameter of the circle. The circle has diameter $34 cm$, so the square has side length $34 \sin{45} =17\sqrt{2} cm$

Do I now use trig to find the last missing side knowing that side length plus the $3m$ length given and the diagonal known. Multiply those together and convert to the right units before I do so?

What last missing side?
You have the dimensions of a square prism; 3m x 17√2 cm x 17√2 cm
You just need to find how many times greater that is than 1" x 1' x 1'
Convert to common units then divide one by the other

You needn't have resorted to trig to find the squares side, Pythagoras would've done.

 

[youngstudent16]

I'n not so sure that the question implies the rectangle needs a dimension of 1ft. the board foot is just a volume, the shape can vary.
What last missing side?
You have the dimensions of a square prism; 3m x 17/√2 cm x 17/√2 cm
You just need to find how many times greater that is than 1" x 1' x 1'
Convert to common units then divide one by the other

You needn't have resorted to trig to find the squares side, Pythagoras would've done.

The correct answer was 4.6 I got something different using those numbers

 

[billy_joule]

I think the books answer is wrong.
1" x 1' x 1' ≈ 2.36 litres
3m x 17√2 cm x 17√2 cm ≈ 173 litres

However you interpret the question I don't think you could get 4.6 as an answer

http://www.wolframalpha.com/input/?i=(3m+*+17√2+cm+*+17√2+cm)+/+board+foot

 

[youngstudent16]

I think the books answer is wrong.
1" x 1' x 1' ≈ 2.36 litres
3m x 17√2 cm x 17√2 cm ≈ 173 litres

However you interpret the question I don't think you could get 4.6 as an answer

http://www.wolframalpha.com/input/?i=(3m+*+17√2+cm+*+17√2+cm)+/+board+foot

It was an online homework computer might have been off or the approach was wrong

 

[William White]

I'n not so sure that the question implies the rectangle needs a dimension of 1ft. the board foot is just a volume, the shape can vary.

are you sure?

you could just have LOTS (about 72!) of tiny little cubes cut from the biggest prism and they would would then not be board feet...

if the shape was not important, why the need to make the prism? you'd use every inch of the wood.

 

[billy_joule]

are you sure?

you could just have LOTS (about 72!) of tiny little cubes cut from the biggest prism and they would would then not be board feet...

If the board foot volume only applied to boards with dimensions of 1" x 1' x 1' what use would it be?
A cube with a board foot volume isn't much less useful than a board foot volume with 1" x 1' x 1' dimension. One might make a nice bowl, the other a chopping board but you'll find neither on the rack at a lumber yard. They'd be in the offcuts bin.
If you do go to a lumber yard, you will find more lengths with the cross sectional dimensions of the cube (~5" x ~5") than the 1" x 1',
The later is rarely used in construction, the former is common for piles, fence posts, retaining walls etc,
You'd pay the same volumetric rate for both; $X/board foot. That's the point of a board foot - consistent pricing regardless of shape. The board foot definition is simple for convenience rather than any correlation to the shape and size of the timber it is used to measure.

Either way I guess we'll never know the intended answer. The given answer is wildly wrong, If the log was machined down to a single plank with dimensions 1" x 1' x 3m, that single board would have over double the board feet as the given answer.