Question

The quantity of net squared timber that can be obtained from a circular log will be around ______________.

• A. 36.4 % of the volume of the log
• B. 78.50 % of the volume of the log
• C. 21.50 % of the volume of the log
• D. 63.6 % of the volume of the log

Hint:

Remember the ratio $2/\pi \approx 0.636$. It is a constant value for any circular log being converted to the largest possible inscribed square.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This is a geometry problem concerning the efficiency of converting a cylindrical log into a square beam (net squared timber).

Step 2: Key Formula or Approach:
1. Cross-sectional area of circular log = $\frac{\pi D^2}{4}$.
2. The largest square that can be inscribed has a diagonal equal to diameter $D$.
3. Side of the square ($s$) = $\frac{D}{\sqrt{2}}$.
4. Area of square timber = $s^2 = \frac{D^2}{2}$.

Step 3: Detailed Explanation:

• We calculate the ratio of the square area to the circular area to find the volume percentage.
\[ \text{Efficiency} = \frac{\text{Area of Square}}{\text{Area of Circle}} = \frac{\frac{D^2}{2}}{\frac{\pi D^2}{4}} \]

• Simplify the ratio:
\[ \text{Efficiency} = \frac{D^2}{2} \times \frac{4}{\pi D^2} = \frac{2}{\pi} \]

• Numerical Calculation:
\[ \frac{2}{3.14159} \approx 0.6366 \]

• Percentage = $0.6366 \times 100 = 63.66\%$.

• This implies that approximately 63.6% of the log's total volume can be recovered as the primary square beam, while the remaining ~36.4% is lost as "slabs" or "waste."

Step 4: Final Answer:
The quantity of net squared timber is 63.6 % of the volume of the log.