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Examoogle • User • info@examoogle.com • EE-2024-8821
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In which is following equation for find out horizontal distance in trigonometric leveling when base of object is inaccessible and nearest station is high to away station.
D=b+hdcotα2tanα2tanα1−tanα2
D=b−hdcotα2tanα2tanα1−tanα2
D=btanα2−hdtanα1−tanα2
Both B and C
Both B and C
When the base of an object is inaccessible and the instrument station nearer to the object is at a higher elevation than the farther station, the horizontal distance D is given by D=tanα1−tanα2(b−hdcotα2)tanα2. Expanding this expression yields D=tanα1−tanα2btanα2−hd, making both Option B and Option C mathematically equivalent and correct.
When the base of an object is inaccessible and the instrument station nearer to the object is at a higher elevation than the farther station, the horizontal distance D is given by D=tanα1−tanα2(b−hdcotα2)tanα2. Expanding this expression yields D=tanα1−tanα2btanα2−hd, making both Option B and Option C mathematically equivalent and correct.
D=tanα1−tanα2(b−hdcotα2)tanα2 — Horizontal distance formula when near station is higher
D=tanα1−tanα2btanα2−hd — Simplified equivalent expression for horizontal distance
In trigonometrical levelling, when instrument stations are at different levels, the height difference hd between the two instrument axes affects the geometry. By setting up simultaneous equations using the vertical angles α1 and α2 measured from the near and far stations respectively, the horizontal distance D is solved by eliminating the vertical height of the object above the instrument axis.
α1 is the angle of elevation measured from the near station A.
α2 is the angle of elevation measured from the far station B.
b is the horizontal distance between instrument stations A and B.
hd is the difference in height between the instrument axes at stations A and B.
Allows accurate distance measurement without direct access to the object base
Accounts for level differences between instrument stations
Requires precise measurement of vertical angles and inter-station distance
Sensitive to atmospheric refraction errors over long distances
Determining heights of inaccessible structures like towers and chimneys
Topographical surveying in hilly or uneven terrain
Option A represents the case where the nearer instrument station is at a LOWER elevation than the farther station.
Multiplying (b−hdcotα2) by tanα2 gives btanα2−hdcotα2tanα2=btanα2−hd, showing why B and C are identical.
D is correct — Both Option B and Option C represent the exact same mathematical formula for horizontal distance when the near station is higher than the far station.
Remember that if the near station is LOWER, the sign of hd changes to positive (b+hdcotα2). If HIGHER, the sign is negative (b−hdcotα2).