Question

10 formula 0.5 points the angle of depression from the top of a cliff to a nearby town is 18 degrees. if the top of the cliff is 360 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\ with your response. answer

Explanation:

Step1: Understand the trigonometric relationship

The angle of depression is equal to the angle of elevation from the town to the top of the cliff (alternate interior angles). We have a right triangle where the height of the cliff (opposite side to the angle of elevation) is 360 feet, and we need to find the adjacent side (distance from town to base of cliff), let's call it \( x \). We use the tangent function, which is \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). Here, \( \theta = 18^\circ \), opposite = 360, adjacent = \( x \). So \( \tan(18^\circ)=\frac{360}{x} \).

Step2: Solve for \( x \)

Rearranging the formula for \( x \), we get \( x = \frac{360}{\tan(18^\circ)} \). Now, calculate \( \tan(18^\circ) \) using a calculator in degree mode. \( \tan(18^\circ)\approx0.3249 \). Then \( x=\frac{360}{0.3249}\approx1107.9 \).

Answer:

1107.9