Problem 33 - A Graphical Approach to Precalculus with Limits

Question

From a window 30.0 feet above the street, the angle of elevation to the top of the building across the street is \(50.0^{\circ}\) and the angle of depression to the base of this building is \(20.0^{\circ} .\) Find the height of the building across the street.

Step-by-Step Solution

Answer

The building height is approximately 101.4 feet.

Step 1: Draw the Situation and Define Variables

Start by sketching the situation. You have a point (the window) 30 feet above the street. The angle of elevation from this point to the top of the building is \(50^{\circ}\), and the angle of depression to the base of the building is \(20^{\circ}\). Define \(d\) as the distance from the base of the building to the point directly below the window on the street.

Step 2: Apply Trigonometry to Find Distance

Use the tangent function, which relates angles in a right triangle to opposite and adjacent sides.For the base of the building:\[ \tan(20^{\circ}) = \frac{30}{d} \]Solve for \(d\):\[ d = \frac{30}{\tan(20^{\circ})} \]

Step 3: Find Additional Height from Top of Window to Top of Building

For the top of the building, use the tangent function again:\[ \tan(50^{\circ}) = \frac{h'}{d} \]where \(h'\) is the additional height from the top of the window to the top of the building.Solve for \(h'\):\[ h' = d \times \tan(50^{\circ}) \]

Step 4: Calculate Total Height of Building

The total height of the building is the height from the street to the window plus \(h'\):\[ \text{Height} = 30 + h' \]Substitute \(d\) and \(h'\) from Steps 2 and 3 to find the final height.

Key Concepts

Understanding the Angle of Elevation

The angle of elevation is a crucial concept in trigonometry. It is the angle formed between the horizontal line of sight and the line from the observer to a higher point. Imagine standing on the ground and looking up to the top of a nearby building. The angle your line of sight makes with the horizontal line is the angle of elevation. Here are some key things to remember about the angle of elevation:

• It is measured from the horizontal upward.
• It helps in determining the height of objects that are above the observer.
• In our problem, the angle of elevation is given as \(50^{\circ}\).

This angle is vital because it allows us to use trigonometric functions, like tangent, to calculate distances and heights. Remember, the greater the angle of elevation, the steeper the line of sight will be.

Exploring the Angle of Depression

While the angle of elevation deals with looking up, the angle of depression involves looking down. This angle is the one formed between the horizontal line and the observer's line of sight down to a lower point. If you're looking out from a window down to the base of a building, you are dealing with the angle of depression. Here are some points to keep in mind:

• This angle is always measured from the horizontal downwards.
• It is helpful in determining how far things are below the staring point.
• In the problem, the angle of depression given is \(20^{\circ}\).

Just like the angle of elevation, the angle of depression can also be used with trigonometric functions to calculate relevant distances or heights. It helps us understand how steep a downward line of sight is.

Utilizing the Tangent Function

The tangent function is a fundamental tool in trigonometry used to solve problems involving right triangles. It relates an angle of a right triangle to the ratio of the length of the opposite side to the adjacent side. When you hear 'tangent function', think of this simple formula:\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]Key details about the tangent function:

• \(\theta\) is the angle you’re focusing on.
• The 'opposite' is the side opposite to \(\theta\).
• The 'adjacent' is the side next to \(\theta\) that is not the hypotenuse.

In our specific problem, tangent allows us to find distances and additional height by rearranging this formula. It's all about using known sides and angles to unlock unknown values that help in solving problems.

Characteristics of a Right Triangle

A right triangle is a triangle that has one angle measuring \(90^{\circ}\). This feature is the foundation allowing us to use trigonometric ratios like tangent to solve problems. In any context of angles of elevation and depression, right triangles naturally emerge. Here's what makes them special:

• One angle is exactly \(90^{\circ}\), known as the right angle.
• The side opposite this angle is known as the hypotenuse, the longest side.
• The other two sides are referred to as the adjacent and the opposite, depending on which angle you are considering.

In real-world applications, right triangles become very useful in calculating heights and distances without direct measurement, especially using the tangent function just as we saw with our exercise solution. Identifying the right triangle in the problem setup will guide you in applying the correct trigonometric approach.