Measuring the radio mast with a clinometer
A surveying team from a school in Otjozondjupa sets out to measure the height of a radio mast standing on level ground. Two points P and Q are marked on a straight line through the base of the mast, with Q farther from the mast and PQ = 40 m.
At P a learner holds a clinometer 1.5 m above the ground and measures the angle of elevation of the top of the mast as 41°; at Q, with the clinometer at the same height, the angle is 28°.
The team's draft report states: 'Height of mast = 54.8 m. The 28° angle is accurate to within ±1°.' The teacher wants the height statement defended or corrected, and also needs the length of a guy wire that will run from the top of the mast to an anchor on level ground, making 55° with the horizontal.
1. Derive an expression for the height of the mast above clinometer level and use it to find the height of the mast above the ground. Comment on the report's value of 54.8 m.
Read the worked answer
Let h = height of the mast above clinometer level and x = horizontal distance from P to the base of the mast.
From P: tan 41° = h/x, so x = h cot 41°.
From Q: tan 28° = h/(x + 40), so x + 40 = h cot 28°.
Subtracting: 40 = h(cot 28° − cot 41°).
cot 28° = 1 ÷ 0.5317 = 1.8807; cot 41° = 1 ÷ 0.8693 = 1.1504; difference = 0.7303.
h = 40 ÷ 0.7303 = 54.77 m.
Height above ground = 54.77 + 1.50 = 56.27 m ≈ 56.3 m.
The report's 54.8 m is the height above the clinometer, not above the ground: the 1.5 m instrument height must be added. This is the key judgement - angles of elevation measured by an observer give heights above the observer's eye, and the offset is only negligible if it is small compared with the answer.
2. Recalculate the mast height for angles of 27° and 29° at Q and use the results to advise on how the height should be reported.
Read the worked answer
Using 27°: cot 27° = 1 ÷ 0.5095 = 1.9626; difference = 1.9626 − 1.1504 = 0.8122; h = 40 ÷ 0.8122 = 49.25 m; height = 50.75 m ≈ 50.8 m.
Using 29°: cot 29° = 1 ÷ 0.5543 = 1.8040; difference = 1.8040 − 1.1504 = 0.6536; h = 40 ÷ 0.6536 = 61.19 m; height = 62.69 m ≈ 62.7 m.
A ±1° error in the far angle moves the answer from about 50.8 m to 62.7 m - a range of nearly 12 m, roughly 20% of the value.
Why: the calculation rests on the difference of two similar cotangents (41° − 28° = 13°), and the 28° angle appears only through that small difference, so its error is amplified.
Advice: 56.27 m (and even 54.8 m) is false precision. A defensible statement is 'the mast is about 56 m high, plausibly between about 51 m and 63 m given the stated accuracy'. Better practice would be to move Q much farther away or to measure the horizontal distance directly with a tape.
3. Find the length of a guy wire running from the top of the mast to a ground anchor, given that the wire makes 55° with the horizontal, and state how far from the base the anchor must be placed.
Read the worked answer
Use the right triangle formed by the mast (56.27 m vertical), the ground, and the wire as hypotenuse.
sin 55° = 56.27 ÷ L, so L = 56.27 ÷ 0.8192 = 68.7 m.
tan 55° = 56.27 ÷ d, so d = 56.27 ÷ 1.4281 = 39.4 m from the base of the mast.
Check: 39.4² + 56.27² = 1 552 + 3 166 = 4 718, and √4 718 = 68.7 ✔.
Judgement: the 55° is only valid if the anchor ground is at the same level as the base; if the ground falls away, the same angle no longer corresponds to that vertical height and the design must use the difference in ground levels. Also 68.7 m is a theoretical minimum - extra wire is needed for terminations and for sag, so the order should exceed the calculated length.