Question

A mountaintop is a height y above the level ground. A woman measures the angle of elevation of the
mountaintop to be θ when she is a horizontal distance x from the mountaintop. After walking a
distance d closer to the mountain, she measures the angle of elevation of the mountaintop to be φ.
Neglecting the height of the woman’s eyes above the ground, draw a well-labelled diagram
representing this situation and find an expression for the height of the mountain, y, in terms of d, φ,
and θ. Note that your expression cannot contain x.

Answer 1

Refer to the diagram shown below.
We want to find y in terms of d, φ and θ.

By definition,
$tan (\theta) = \frac{y}{x} \\\\ tan( \phi) = \frac{y}{x-d}$

Therefore
y = x tan(θ)                   (1)
y = (x - d) tan(φ)           (2)

Equate (1) and (2).
$(x - d) \, tan(\phi) = x \, tan(\theta) \\ x[tan(\phi) - tan(\theta)] = d \, tan(\phi) \\ x= \frac{d tan(\phi)}{tan(\phi)-tan(\theta)}$

From (1), obtain the required expression for y.

Answer:
$y= \frac{d \, tan(\phi) \, tan(\theta)}{tan(\phi)-tan(\theta)}$