Question

The volume of the log estimated using the Quarter girth formula is ______________.

• A. 78.50 % of the actual volume
• B. 21.50 % of the actual volume
• C. 78.50 % more than the actual volume
• D. 21.50 % more than the actual volume

Hint:

Timber traders prefer this formula because it automatically deducts about 21.5% of the volume, which represents the wood lost when converting a round log into a square beam.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The Quarter Girth formula is a traditional method used by timber traders to estimate the usable volume of a log, accounting for wastage during squaring. The question asks for its mathematical relationship with the actual volume of a cylinder.

Step 2: Key Formula or Approach:
1. Actual Volume of a cylinder ($V_a$) = $\frac{g^2}{4\pi} \times L$.
2. Quarter Girth Volume ($V_{qg}$) = $(\frac{g}{4})^2 \times L = \frac{g^2}{16} \times L$.
Where $g$ is the girth and $L$ is the length.

Step 3: Detailed Explanation:

• Let us compare the two volumes by taking their ratio:
\[ \text{Ratio} = \frac{V_{qg}}{V_a} = \frac{\frac{g^2}{16} \times L}{\frac{g^2}{4\pi} \times L} \]

• Simplify the expression:
\[ \text{Ratio} = \frac{4\pi}{16} = \frac{\pi}{4} \]

• Using $\pi \approx 3.14159$:
\[ \text{Ratio} \approx \frac{3.14159}{4} = 0.78539 \]

• Converting to percentage:
\[ 0.7854 \times 100 = 78.54\% \]

• This means the Quarter Girth formula provides only about 78.5% of the true geometric volume. The remaining 21.5% is considered "wastage" or the "bark allowance" in commercial trade.

Step 4: Final Answer:
The Quarter girth volume is 78.50 % of the actual volume.