Problem 50 - Algebra and Trigonometry

Question

A flagpole is situated on top of a building. The angle of elevation from a point on level ground 330 feet from the building to the top of the flagpole is \(63^{\circ} .\) The angle of elevation from the same point to the bottom of the flagpole is \(53^{\circ} .\) Find the height of the flagpole to the nearest tenth of a foot.

Step-by-Step Solution

Answer

Apply the respective tangent values, and compute to get the height of the flagpole to the nearest tenth of a foot.

Step 1: Define the problem with a diagram

Start by visualizing the problem with a simple diagram. You can draw two right triangles (one for the building and one for the flagpole on top). For the triangle formed by the 63-degree angle (the top of the flagpole to the ground), let's denote its opposite side (the total height from the ground to the top of the flagpole) as \(b\). For the triangle formed by the 53-degree angle (the bottom of the flagpole to the ground), let's denote its opposite side (the height from the ground to the bottom of the flagpole) as \(a\). You are required to find the height of the flagpole, which is \(b - a\).

Step 2: Use trigonometric relationships to set up equations

From a trigonometric relationship, we know that the tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Therefore, we can express \(a\) and \(b\) in terms of the given distances and angles as follows: \[a = 330 \cdot \tan(53^{\circ})\] and \[b = 330 \cdot \tan(63^{\circ})\].

Step 3: Apply subtraction to find the height of the flagpole

Subtract \(a\) from \(b\) to find the height of the flagpole. This can be written as: \[b - a = 330 \cdot \tan(63^{\circ}) - 330 \cdot \tan(53^{\circ})\]