Question 2 of 7: True/False Statements with Corrections

Question 2: True/False Statements with Corrections (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Ten statements spanning GNSS height systems, geodetic surface relationships, horizontal- and vertical-curve stationing, bearing/azimuth reversal, traverse classification, and random-error propagation.

Find. A True/False verdict for each statement, with the correcting statement supplied for every false one.

Approach. Judge each statement against the governing definition or formula; where it is false, state the smallest correction that makes it true. The two quantitative statements (4 and 10) are settled by direct computation.

1. Statement 1 — F. Three satellites determine only the three position unknowns and leave the receiver-clock bias unsolved. Correction: a minimum of four satellites is required — three for the $X,Y,Z$ position and a fourth to solve the receiver-clock error.
2. Statement 2 — F. GPS observes geometric range to the satellites, so it delivers heights referred to the reference ellipsoid. Correction: GPS heights are ellipsoidal heights $h$ (relative to the ellipsoid), not orthometric heights referred to the geoid; a geoid model $N$ is needed to convert them.
3. Statement 3 — F. The correct relation among the three surfaces is $h = H + N$. Correction: the ellipsoidal height $h$ equals the orthometric height $H$ plus the geoidal height (undulation) $N$ — not "geoidal height $=$ ellipsoidal $+$ orthometric."
4. Statement 4 — F. By the chord definition, $D_c = 2\arcsin\!\left(\dfrac{50}{R}\right) = 2\arcsin\!\left(\dfrac{50}{900}\right) = 6^\circ22'10''$, whereas by the arc definition $D_a = \dfrac{5729.578}{R} = \dfrac{5729.578}{900} = 6^\circ21'58''$. The printed value is the arc-definition result. Correction: $6^\circ21'58''$ is the degree of curve by the arc definition; the chord-definition value is $\boxed{6^\circ22'10''}$.
5. Statement 5 — T. Azimuth $235^\circ$ lies in the third quadrant, so its bearing is $S(235^\circ-180^\circ)W = S55^\circ W$ ✓; the back azimuth is $235^\circ-180^\circ = 55^\circ$, whose bearing is $N55^\circ E$ ✓. The statement is consistent throughout.
6. Statement 6 — F. For a parabola the tangent (vertical) offset from the tangent grows with the square of the distance. Correction: the tangent offsets vary as the square of the distance from the point of tangency ($y \propto x^2$), not linearly with distance.
7. Statement 7 — F. The first half is right (PC $=$ PI $-\,T$) but the second is wrong: the PT is reached from the PC along the arc, not out to the PI and back. Correction: the station of the PT equals the station of the PC plus the curve length $L$ (PT $=$ PC $+\,L$), which differs from PI $+\,T$ because $L \neq 2T$.
8. Statement 8 — F. An equal-tangent vertical curve is symmetric about the PVI, so each end lies half the length away. Correction: PVC $=$ PVI $-\,L/2$ and PVT $=$ PVI $+\,L/2$ (half the curve length, not the full length).
9. Statement 9 — F. A traverse that begins on one known point and ends on a different known point is still closed (a closed connecting or link traverse), because it can be checked against control. Correction: such a traverse is a closed (connecting/link) traverse; an open traverse is one that ends at a point of unknown position with no closure check.
10. Statement 10 — T. Random (accidental) errors propagate as the root-sum-square, so for $n$ equal measurements the total is $E = e\sqrt{n} = 0.006\sqrt{36} = 0.006(6) = \boxed{\pm0.036\ \text{m}}$. The statement is correct.