Question

The volume of the log having 3 m length with 90 cm and 75 cm thick and thin end diameter respectively is ______________.

• A. 0.128 m$^3$
• B. 1.260 m$^3$
• C. 2.10 m$^3$
• D. 1.100 m$^3$

Hint:

In exams, if the standard geometric volume does not match perfectly, check if the Quarter Girth formula $(g/4)^2 \times L$ was used, as it is standard in Indian timber trade.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the volume of a tapering log (frustum of a cone). Standard forestry practices often use Smalian's or Huber's formula for such calculations.

Step 2: Key Formula or Approach:
Using Smalian's formula: \( V = \frac{A_1 + A_2}{2} \times L \).
where \( A = \frac{\pi d^2}{4} \) is the cross-sectional area at each end.

Step 3: Detailed Explanation:

• Convert diameters to meters: $d_1 = 0.90$ m, $d_2 = 0.75$ m. Length $L = 3$ m.

• Calculate Area 1 (thick end):
\[ A_1 = \frac{\pi \times 0.90^2}{4} = 0.7854 \times 0.81 \approx 0.636 \text{ m}^2 \]

• Calculate Area 2 (thin end):
\[ A_2 = \frac{\pi \times 0.75^2}{4} = 0.7854 \times 0.5625 \approx 0.442 \text{ m}^2 \]

• Calculate average area:
\[ A_{avg} = \frac{0.636 + 0.442}{2} = 0.539 \text{ m}^2 \]

• Calculate Volume ($V$):
\[ V = 0.539 \times 3 = 1.617 \text{ m}^3 \]

• Note: If using the Quarter Girth formula commercially ($g = \pi d$):
\[ g_{avg} = \pi \times \frac{0.90+0.75}{2} = 3.1416 \times 0.825 \approx 2.59 \text{ m} \]
\[ V_{qg} = (\frac{2.59}{4})^2 \times 3 = 0.419 \times 3 = 1.258 \text{ m}^3 \]

• The option 1.260 m$^3$ matches the commercial Quarter Girth calculation perfectly.

Step 4: Final Answer:
The volume of the log is 1.260 m$^3$.