Problem 41 - Algebra and Trigonometry

Question

The angle of inclination from the base of the John Hancock Center to the top of the main structure of the Willis Tower is approximately \(10.3^{\circ}\). If the main structure of the Willis Tower is 1451 feet tall, how far apart are the two skyscrapers? Assume the bases of the two buildings are at the same elevation.

Step-by-Step Solution

Answer

The two skyscrapers are approximately 7973 feet apart.

Step 1: Identify the Given Information

The angle of inclination from the base of the John Hancock Center to the top of the main structure of the Willis Tower is given as \(10.3^{\text{°}}\). The height of the Willis Tower is provided as 1451 feet.

Step 2: Understand the Problem

To find the distance between the two skyscrapers, consider the scenario as a right triangle where the height of the Willis Tower forms the opposite side to the angle of inclination. The distance we need to find is the length of the adjacent side.

Step 3: Use the Tangent Function

In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. The formula is: \[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]. Here, \(\theta = 10.3^{\text{°}}\) and opposite side = 1451 feet.

Step 4: Set Up the Equation

From the tangent function, set up the equation: \[\tan(10.3^{\text{°}}) = \frac{1451}{\text{distance}}\]

Step 5: Solve for the Distance

Rearrange to solve for the distance: \[\text{distance} = \frac{1451}{\tan(10.3^{\text{°}})}\]. Calculate the value using a calculator: \(\tan(10.3^{\text{°}})\) approximately equals 0.1820. Thus, distance = \( \frac{1451}{0.1820} \approx 7973 \text{ feet}\).

Key Concepts

Angle of Inclination

The angle of inclination is a measure of how steep a line is relative to the horizontal. In the context of the given exercise, the angle of inclination is given as \(10.3^{\text{°}}\). This angle forms between the horizontal line connecting the bases of the two skyscrapers and the line from the base of one skyscraper (John Hancock Center) to the top of the Willis Tower. Understanding the angle of inclination is crucial in problems involving slopes, heights, and distances. It helps you relate a horizontal change (distance between buildings in this case) to a vertical change (height of the Willis Tower). By knowing the angle of inclination and one side of the right triangle, you can determine other dimensions within the triangle using trigonometric functions.

Tangent Function

The tangent function in trigonometry is a key concept that relates the angles of a right triangle to the ratios of its sides. Specifically, the tangent of an angle (denoted as \( \tan(\theta) \)) in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. This is mathematically expressed as:
\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]
In the provided exercise, the angle is \(10.3^{\text{°}}\), the opposite side is the height of the Willis Tower (1451 feet), and the adjacent side is the horizontal distance between the two skyscrapers. By rearranging the tangent formula to solve for the adjacent side, we get:
\[\text{adjacent} = \frac{\text{opposite}}{\tan(\theta)}\]
Therefore, you can calculate the distance between the two skyscrapers using this rearranged formula and a calculator.

Right Triangle

A right triangle is a triangle with one angle measuring exactly 90 degrees. It has three sides:

• The hypotenuse (the longest side opposite the right angle)
• The opposite side (the side opposite the angle in question)
• The adjacent side (the side adjacent to the angle in question)

In the exercise given, the scenario can be visualized as a right triangle where:

• The height of the Willis Tower (1451 feet) is the length of the opposite side
• The angle of inclination (10.3°) is used to determine the tangent value
• The distance between the two skyscrapers is the adjacent side

Using these definitions and the tangent function, you can solve for the unknown distance by carefully setting up and solving the trigonometric equations in a step-by-step manner.