Question

1. an observer standing on the top of a vertical cliff spots a house in the adjacent valley at an angle of depression of 12°. the cliff is 60m tall. how far is the house from the base of the cliff?

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the height of the cliff (60 m) is the opposite side to the angle of depression (12°), and the distance from the house to the base of the cliff (x) is the adjacent side. We use the tangent function: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$.

Step2: Set up the equation

$\tan(12^\circ) = \frac{60}{x}$

Step3: Solve for x

$x = \frac{60}{\tan(12^\circ)}$
Calculate $\tan(12^\circ) \approx 0.2126$
$x \approx \frac{60}{0.2126} \approx 282.2$

Answer:

The house is approximately 282.2 meters from the base of the cliff.