Question

Match List I with List II
List I: A. Net squared timber from log, B. Quarter girth formula, C. Bottom portion of tree, D. Middle portion of tree
List II: I. Frustrum of cone, II. Frustrum of neiloid, III. 63.6 % of the actual volume, IV. 78.50 % of the actual volume, V. Frustrum of paraboloid

• A. A‐IV , B‐III , C‐ II, D‐ V
• B. A‐IV , B‐II , C‐ III, D‐ I
• C. A‐III , B‐IV , C‐I, D‐ V
• D. A‐III , B‐IV , C‐ II, D‐ V

Hint:

Geometric shapes of tree stems from base to tip: Neiloid (base) \(\rightarrow\) Paraboloid (middle) \(\rightarrow\) Cone (top). Mnemonic: NPC (like a computer character). Quarter girth is always ~78.5% of actual.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question involves matching tree stem geometry and timber conversion constants with their respective mathematical descriptions. Tree stems are not perfect cylinders but resemble various solids of revolution.

Step 2: Detailed Explanation:

• Net Squared Timber (A): When a round log is converted into a square beam (cant), there is a significant loss in volume. Geometrically, the maximum square that can be cut from a circle has an area about 63.6% (III) of the circle's area. In forestry practice, the "net squared timber" refers to this recovery ratio.

• Quarter Girth Formula (B): Also known as the Hoppus rule (\(V = (g/4)^2 \times L\)), it is widely used in the timber trade. It underestimates the volume compared to the actual cylinder. The volume calculated by this formula is approximately 78.5% (IV) of the actual geometrical volume (\( \pi \times r^2 \times L\)).

• Stem Forms: The shape of a tree trunk varies from the base to the top.
- Bottom Portion (C): Near the ground, due to buttressing and rapid taper, the shape resembles a frustrum of a neiloid (II).
- Middle Portion (D): The main merchantable part of the trunk usually tapers more gradually and follows the shape of a frustrum of a paraboloid (V).
- Top Portion: Usually resembles a frustrum of a cone or a simple cone.

Step 3: Final Answer:
The correct matching is: A-III, B-IV, C-II, D-V.