Question

5 formula 2 points the angle of depression from the top of a cliff to a nearby town is 13 degrees. if the top of the cliff is 217 feet above the town, how far is the town from the base of the cliff? cliff town be sure your calculator is in deg mode, and use the proper trig function on your calculator in the computation. round your answer to the nearest tenth of a foot, but do not include \ft\ or \feet\ in your response.

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the opposite side to the angle of depression (which is equal to the angle of elevation from the town to the top of the cliff) is the height of the cliff (217 feet), and the adjacent side is the distance from the town to the base of the cliff (let's call it \(x\)). We use the tangent function: \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 13^\circ\), opposite \(= 217\), adjacent \(= x\). So \(\tan(13^\circ)=\frac{217}{x}\).

Step2: Solve for \(x\)

Rearrange the formula to solve for \(x\): \(x=\frac{217}{\tan(13^\circ)}\).

Step3: Calculate the value

Using a calculator in degree mode, \(\tan(13^\circ)\approx0.230868\). Then \(x = \frac{217}{0.230868}\approx939.9\).

Answer:

939.9