Problem 48 - Algebra and Trigonometry with Analytic Geometry

Question

From a point \(A\) that is \(8.20\) meters above level ground, the angle of elevation of the top of a building is \(31^{\circ} 20^{\prime}\) and the angle of depression of the base of the building is \(12^{\circ} 50^{\prime}\). Approximate the height of the building.

Step-by-Step Solution

Answer

The height of the building is the sum of \(BC\) and \(8.20\) meters.

Step 1: Understand the Setup

We have a point \(A\) which is \(8.20\) meters off the ground. The angles of elevation and depression to the building's top and base are given. The angle of elevation \(\angle BAC\) is \(31^{\circ} 20^{\prime}\) and the angle of depression \(\angle BAD\) is \(12^{\circ} 50^{\prime}\). We need to find the height of the building, essentially the vertical distance from the point where the base of the building meets the ground to the building's top.

Step 2: Use Information about the Angles

The angle of elevation is the angle formed with the horizontal where the object of interest is above the line of sight. The angle of depression is formed below the horizontal line of sight. This means the line of sight towards the base of the building is below the horizontal line from point \(A\) and forms a separate right triangle.

Step 3: Calculate the Distance to the Building

Let's denote the point where the top of the building intersects the line of sight from \(A\) as \(C\) and the base as \(D\). Using the tangent function from trigonometry, for the triangle \(AHD\) with base \(D\) and height \(8.20\), we have:\[tan(12^{\circ} 50^{\prime}) = \frac{HD}{AD}\]From this equation, we solve for \(AD\) where \(AD\) is the horizontal distance from \(A\) to the building's base.

Step 4: Solve for Horizontal Distance

Given our equation in Step 3, solving it gives:\[AD = \frac{8.20}{\tan(12^{\circ} 50^{\prime})}\]Compute \(AD\) using the tangent value for \(12^{\circ} 50^{\prime}\).

Step 5: Use Optimal Trigonometric Functions

Now to find \(AB\) where \(B\) is at the top of the building, use:\[tan(31^{\circ} 20^{\prime}) = \frac{BC}{AD}\]Where \(BC\) is the height from \(A's\) horizontal line of sight to the top of the building.

Step 6: Solve for Building Height

Given \(BC = AD \cdot \tan(31^{\circ} 20^{\prime})\), compute this using the previously found \(AD\) and then finally add \(8.20\) meters to \(BC\) to get the total height of the building.

Key Concepts

Angle of Elevation

An angle of elevation refers to the angle formed between the line of sight and the horizontal plane when you are looking up at an object. In trigonometry, it is often used to find the height of an object using a right triangle. Imagine standing on the ground, looking up at the top of a building. The angle that your line of sight makes with the horizontal is the angle of elevation.

• It's measured from the observer's eye level to the object above.
• In problems involving tall structures, it's crucial for determining unknown distances or heights.
• Trigonometric functions like the tangent are frequently used in these calculations.

In our exercise, with point A at 8.20 meters above the ground, the angle of elevation is given as \(31^{\circ} 20'\). This helps us understand how steeply we are looking up to see the top of the building from point A.

Angle of Depression

The angle of depression is commonly misunderstood, yet it's an essential concept in trigonometry, particularly when observing objects below the horizontal plane from an elevated position. This angle is formed between the horizontal line and the observer's line of sight down to an object.

• It is measured downward from the observer's horizontal line of sight.
• Used to calculate distances between heights and lower points in relation to each other.
• Often involves applying trigonometric functions like sine, cosine, or tangent.

In this exercise, the angle of depression from point A to the base of the building is \(12^{\circ} 50'\). This angle helps in setting up the right triangle used to find horizontal and vertical distances from point A.

Tangent Function

The tangent function is a fundamental trigonometric function used extensively in problems involving right triangles. It is defined as the ratio of the opposite side to the adjacent side within a right triangle.

• Represented as \(\tan(\theta) = \frac{\text{Opposite Side}}{\text{Adjacent Side}}\).
• It's useful for finding angles or side lengths in right triangles.
• In elevated/depressed scenarios, it helps translate between height and distance.

In our example, tangent is used to relate the heights viewed from point A (either up or down). We use \(\tan(12^{\circ} 50') = \frac{HD}{AD}\) to find the distance from A to the building's base. Similarly, \(\tan(31^{\circ} 20') = \frac{BC}{AD}\) helps find the distance from A to the top.

Right Triangles

Right triangles are a key component in solving problems about angles of elevation and depression. A right triangle has one angle that's exactly 90 degrees, which simplifies calculations in trigonometric equations since the sides can be linked via functions like sine, cosine, and tangent.

• They consist of an opposite, adjacent side, and the hypotenuse.
• Enable straightforward use of trigonometric ratios to ascertain unknown lengths and angles.
• Provide a simplified model to apply angles like elevation and depression.

In this problem, the right triangles formed with angle \(\angle BAC\) and \(\angle BAD\) are essential. Each triangle helps us use known angles to calculate unknown lengths, including finding the full height of the building by stacking known parts together.