Problem 67 - Precalculus: A Concise Course

Question

You are standing 45 meters from the base of the Empire State Building. You estimate that the angle of elevation to the top of the 86 th floor (the observatory) is \(82^{\circ} .\) If the total height of the building is another 123 meters above the 86 th floor, what is the approximate height of the building? One of your friends is on the 86 th floor. What is the distance between you and your friend?

Step-by-Step Solution

Answer

The approximate height of the Empire State Building is the sum calculated in step 2. The distance between you and your friend is the distance calculated in step 3.

Step 1: Calculate the height to the 86th floor

The tangent of the angle of elevation is equal to the height of the building divided by the distance to the building. So, rearrange the formula to: \( height = \tan(angle) \times distance \). Calculate the height to the 86th floor by multiplying the tangent of 82 degrees by 45 meters.

Step 2: Calculate the total height

The total height of the Empire State Building is the height to the 86th floor plus the total height above the 86th floor. So add the height to the 86th floor calculated in step 1 to the total height above the 86th floor (123 meters).

Step 3: Calculate the distance between you and your friend

Your friend is standing at the 86th floor which is at an angle of 82 degrees from your current position. The distance to your friend can be calculated using the sine function, since: \(\sin(angle) = \frac{distance}{hypotenuse}\), therefore: \(distance = \sin(82^{\circ}) \times hypotenuse\), where the hypotenuse is the distance calculated in step 1.

Key Concepts

Angle of Elevation

When you stand at a distance and look up at a higher point, the angle your line of sight makes with the horizontal line from your eyes is called the "angle of elevation." It is measured in degrees and indicates how steeply you need to look up to see the top of an object. In the problem of the Empire State Building, the angle of elevation is given as 82 degrees.
Understanding this concept is crucial when calculating the height and distance of objects without measuring directly. By knowing this angle along with the distance from the object, other trigonometric calculations can be made to find unknown measurements.

Tangent Function

The tangent function is a trigonometric function that relates angles to the ratio of opposite to adjacent sides in a right triangle. It is commonly used in scenarios involving angles of elevation or depression.

• The tangent of an angle is expressed as: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)
• In our exercise, \( \theta \) is the angle of elevation (82 degrees), the opposite is the height of the building, and the adjacent is the distance from the building (45 meters).

To calculate the height of the 86th floor, we rearrange this to: \( \text{height} = \tan(82^{\circ}) \times 45 \). This equation uses the known distance and angle to solve for the height, which is practical when direct measurement is challenging.

Sine Function

The sine function can also be pivotal in height and distance calculations when working with right triangles. It connects an angle with the ratio of the opposite side to the hypotenuse.

• Expressed as: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)
• In this problem, \( \sin(82^{\circ}) \) helps to find the distance vertically between the observer and their friend by considering the hypotenuse, which was calculated involving the tangent previously.

Using \( \sin(82^{\circ}) \), one can calculate the direct distance between you and your friend, contributing to assessing physical spaces in practical scenarios without prior measurements of the hypotenuse.

Height Calculation

To solve for the height of the beautiful Empire State Building's 86th floor, we need to use the tangent function. With the given angle of elevation and distance, this is straightforward. By inputting the values into the formula \( \text{height} = \tan(82^{\circ}) \times 45 \), we solve for the height to the 86th floor.

Don't forget! The total height of the building includes both this calculated height and the additional 123 meters above the 86th floor. Therefore, add these two numbers together to find out the total height of the building. It's an effective use of trigonometry for real-world applications.

Distance Calculation

Determining the actual distance between two points with one on a higher level requires using trigonometry. By applying the sine function, we can determine the distance between you and your friend on the 86th floor.

• Using a known angle and the hypotenuse calculated from the tangent function, the sine function will find the desired distance.
• Formula application: \( \sin(82^{\circ}) = \frac{\text{distance}}{\text{hypotenuse}} \)

By rearranging to solve for "distance," it's easy to assess how far you'd actually have to travel to reach your friend. It's about understanding how mathematical functions help navigate and solve real-life spatial challenges.