Question

from the top of a 38 metre tall building, i look down to the street below. at an angle of depression of 67 degrees, i see a car driving. five seconds later, it is now at an angle of depression of 22 degrees.
a. is it moving towards or away from the building? show a sketch.

Explanation:

Step1: Recall Angle of Depression Concept

The angle of depression is equal to the angle of elevation from the object to the observer (alternate interior angles). A smaller angle of depression means the object is farther from the base of the building (since $\tan(\theta)=\frac{\text{height}}{\text{horizontal distance}}$, so for a fixed height, smaller $\theta$ implies larger horizontal distance).

Step2: Analyze Initial and Final Angles

Initial angle of depression: $67^\circ$, final angle: $22^\circ$. Since $22^\circ<67^\circ$, the horizontal distance from the building to the car has increased (because $\tan(67^\circ)=\frac{38}{d_1}$ and $\tan(22^\circ)=\frac{38}{d_2}$, with $\tan(67^\circ)>\tan(22^\circ)$ so $d_1 < d_2$).

Step3: Determine Direction of Motion

If the horizontal distance from the building to the car increases over time, the car is moving away from the building.

Sketch Description:

• Draw a vertical line to represent the 38 - metre tall building.
• From the top of the building, draw two horizontal lines (dashed, to represent the line of sight horizontally) and two lines of sight making angles of $67^\circ$ and $22^\circ$ with the horizontal (downwards) to the street.
• Mark the initial position of the car (closer to the building, under the $67^\circ$ line of sight) and the final position (farther from the building, under the $22^\circ$ line of sight).

Answer:

The car is moving away from the building.