In which of the following cases, two instrument stations are used?

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Option D: Base of the object is inaccessible

Explanation & Concepts:
In trigonometrical levelling, two instrument stations are required when the base of the object whose elevation is to be determined is inaccessible. This setup allows the vertical angle to be measured from two different known stations lying in the same or different vertical planes, helping determine the elevation and distance of the object using triangulation principles.

Working Principle: When the base of an elevated object (like a chimney or tower) cannot be reached to measure the horizontal distance directly, two instrument stations ($A$ and $B$) are established at a known distance apart. Angles of elevation to the top of the object are measured from both stations. Using trigonometric relations involving the measured angles and the baseline length, the horizontal distance and relative height are computed.

Key Formulas: $D = \frac{b \cot \alpha_2}{\cot \alpha_1 - \cot \alpha_2}$ — horizontal distance from near station to object when stations are in line; $h = \frac{b \sin \alpha_1 \sin \alpha_2}{\sin(\alpha_1 - \alpha_2)}$ — height of object above the instrument axis

Key Points: Used when direct horizontal distance measurement to the base of an object is impossible.; The two instrument stations must be set up at a precisely measured baseline distance ($b$).; Angles of elevation ($\,\alpha_1\,$ and $\,\alpha_2\,$) are observed from both instrument stations to the top of the object.; Can be solved for both single vertical plane and double vertical plane conditions.

Advantages: Allows determination of heights and distances of inaccessible elevated objects without direct physical measurement.; Highly precise for objects such as church spires, towers, mountain peaks, and chimneys.

Disadvantages: Requires set-up of two instrument stations and accurate measurement of baseline length.; Computations become slightly complex if instrument axes are at different levels.

Applications: Determining heights of tall chimneys, TV towers, and transmission towers.; Topographical survey work in mountainous terrain where physical access is blocked.

Additional Info: Option A is incorrect because infinity is not a practical survey condition.. Option B is incorrect because base accuracy does not dictate instrument station count.. Option C is incorrect because when the base is accessible, only one instrument station is required as distance $D$ can be directly measured with a tape or chain.

Answer: D is correct — Two instrument stations are required in trigonometrical levelling when the base of the target object is inaccessible, preventing direct distance measurement.

Exam Tip: Remember: Single station is sufficient when base is accessible ($h = D 	an \alpha$). Two stations are mandatory when the base is inaccessible.

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Option A: Base of the object is at infinity

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Option B: Base of the object is at accurate position

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Option C: Base of the object is accessible

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