Question

A mountaintop is a height y above the level ground. A woman measures the angle of elevation of the

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