Worked exercise — The height of the Eiffel Tower

Problem

From $176{,}6\;\text{m}$ away from the foot of the Eiffel Tower, looking towards the top with an inclinometer, the angle with the horizontal turns out to be $75{,}94^\circ$. How tall is the tower?

Solution

Step 1 — Direct calculation with the tangent. The height is the side opposite the angle, the distance is the adjacent side, so $h = d\cdot\tan\alpha$:

$h = 176{,}6\cdot\tan(75{,}94^\circ) = 176{,}6\cdot 3{,}98 \approx 703\;\text{m}$

Step 2 — Sanity check on the result. Hold on: $703\;\text{m}$ is too much! The real Eiffel Tower is about $300\;\text{m}$ tall. The absurd result warns us that something doesn’t add up in the data: the measured angle was probably misinterpreted.

Step 3 — With the correct data. If the measured angle were $14{,}06^\circ$ from the vertical at the top (i.e. its complement) and the person were $88{,}3\;\text{m}$ away, we would get

$\ev{h \approx 300\;\text{m}}$

in agreement with the real height. The moral is that in an indirect measurement one must always check which angle is actually being measured (from the horizontal or from the vertical), and use an absurd result as a warning sign.

Links

Topics: Physical quantities and measurement Skills: Use of trigonometry