Question

The Pont du Gard near Nîmes, France, is a Roman aqueduct. An observer in
a hot-air balloon some distance away from the aqueduct determines that the
angle of depression to each end is
54°
54°
and
71°
, respectively. The closest end
of the aqueduct is 270.0 m from the balloon

Answer 1

Let h be the balloon height and let d2 be the horizontal distance from the balloon to the farther end. For an angle of depression θ, tan θ = h / (horizontal distance).

For the closer end (angle 71° and distance 270.0 m):
h = 270.0 · tan 71° ≈ 270.0 · 2.9042 ≈ 784.1 m.

For the farther end (angle 54°):
d2 = h / tan 54° ≈ 784.1 / 1.3763819 ≈ 569.7 m.

So the length of the aqueduct between the two ends (horizontal distance between them) is
d2 − 270.0 ≈ 569.7 − 270.0 ≈ 299.7 m.

Answers:
- Height of the balloon ≈ 784.1 m
- Distance to the farther end ≈ 569.7 m
- Distance between the ends ≈ 299.7 m