CDS Mathematics · Heights and Distances

Observer Above the Ground

An observer above the ground sees the top of a taller object by elevation and its foot by depression; the level line through the eye splits the object in two.

All Heights and Distances notes CDS Mathematics strategy

Why this matters

Nine PYQs, four of them HARD. Draw the horizontal line through the observer's eye. The depression of the foot fixes the ground distance from the observer's own height; the elevation of the top then gives the part ABOVE eye level. Add the two parts.

Concept 1 of 1: Elevation and depression together

From a deck

h

above the water, the foot of a tower is

h

below eye level and its top is some height above. Both share the same ground distance, so one angle fixes the distance and the other gives the rest of the height.

Definition

Observer at height $h$ , depression $\beta$ of the foot: ground distance $d = h\cot\beta$ .
Elevation $\alpha$ of the top: height above eye level $= d\tan\alpha$ . Total $= h + d\tan\alpha$ .
A shorter object below eye level: its top is $d\tan(\text{depression of top})$ below the eye.
A cloud and its reflection: the reflection is as far BELOW the water as the cloud is above.
A line of sight that just touches a dome is tangent to it: distance from the centre $= \dfrac{r}{\sin\theta}$ .

Height of the object

H = h + h\cot\beta\tan\alpha

Worked example

From a window

10

m up, the elevation of a tower's top is

45^\circ

and the depression of its foot is

30^\circ

. Find the tower's height.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2023 · CDS (I) 2023 — Elementary Mathematics · Q55Moderate

Example 1 · Heights and Distances · Observer Above the Ground

A woman is standing on the deck of a ship, which is

h

(in metres) above water level. She observes the angle of elevation of the top of a tower as

60^\circ

and the angle of depression of the base of the tower as

30^\circ

. What is the height of the tower ?

Add the part below eye level

The elevation gives only the part of the tower ABOVE the observer's eye. The observer's own height

h

must be added to get the whole tower.

Drill 8 more on elevation and depression together

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Elevation and depression together
Height of the object
$H = h + h\cot\beta\tan\alpha$

Watch out for (1)

Add the part below eye level
→ Elevation and depression together

Test yourself on Heights and Distances

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Heights and Distances chapter test Drill all 9 questions