Question

If the plane of caliper is interlined by an angle $\theta$ the diameter becomes D sec$\theta$ and error involved is as follows. Select one of the following correct answer.

• A. $\pm$ (D tan$\theta$)/2
• B. $\pm$ (L/D) tan$\theta$ $\times$ 100
• C. $\pm$ D (sec$\theta$ - 1)
• D. D tan$\theta$

Hint:

Because $\sec \theta$ is always greater than or equal to 1, any tilt in the caliper plane will overestimate the true diameter. The error formula is derived simply as: $D_{\text{measured}} - D_{\text{true}} = D\sec\theta - D = D(\sec\theta - 1)$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Tree calipers are used to measure stem diameter at breast height (DBH).
To obtain an accurate measurement, the plane of the caliper must be perpendicular to the longitudinal axis of the tree stem.
If the caliper plane is tilted at an angle $\theta$ relative to the perpendicular plane, an error is introduced.

Step 2: Detailed Explanation:
Let us calculate the error mathematically:
- Let the true perpendicular diameter of the tree stem be $D$.
- If the caliper plane is tilted (inclined) by an angle $\theta$, the measured diameter $D'$ represents the hypotenuse of a right-angled triangle where the true diameter $D$ is the adjacent side:
\[ D' = D \sec \theta \]
- The error introduced in the diameter measurement is the difference between the tilted measurement ($D'$) and the true diameter ($D$):
\[ \text{Error} = D' - D \]
Substitute the value of $D'$ into the equation:
\[ \text{Error} = D \sec \theta - D \]
Factor out $D$ from the terms:
\[ \text{Error} = D (\sec \theta - 1) \]
Depending on the direction of the tilt, this error represents an overestimation, which is written as:
\[ \pm D (\sec \theta - 1) \]
This matches Option (C).

Step 3: Final Answer:
The error involved when the caliper plane is inclined at an angle $\theta$ is $\pm D (\sec \theta - 1)$.