Question

1 an observer stands an unknown distance away from a building that is 80 feet tall. looking up to the top of the building, he notes that the angle of elevation is 52°. determine the distance from the base of the building to the observer. diagram a known distance x 52° option 1 diagram b x known distance 52° option 2 diagram c known distance x 52° option 3 diagram d known distance x° known distance option 4 diagram e known distance

Explanation:

Step1: Analyze the problem

We have a right triangle where the height of the building (opposite side to the angle of elevation) is 80 feet, the angle of elevation is \(52^\circ\), and we need to find the distance from the observer to the base of the building (adjacent side to the angle of elevation), let's call this distance \(x\).

Step2: Identify the trigonometric ratio

We know that in a right triangle, \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 52^\circ\), opposite side \(= 80\) feet, and adjacent side \(= x\). So, \(\tan(52^\circ)=\frac{80}{x}\).

Step3: Solve for \(x\)

We can re - arrange the formula to solve for \(x\): \(x=\frac{80}{\tan(52^\circ)}\). We know that \(\tan(52^\circ)\approx1.2799\). Then \(x = \frac{80}{1.2799}\approx62.5\) feet.

Now, let's analyze the diagrams:

• The building height is the known distance (opposite side), the angle of elevation is \(52^\circ\), and the distance from the observer to the base is \(x\) (adjacent side). So the diagram should have the known distance (building height) as the vertical side, the angle of elevation as \(52^\circ\), and \(x\) as the horizontal side (adjacent side). This matches Diagram C.

Answer:

Option 3 (Diagram C) and the distance is approximately 62.5 feet.