Problem 39 - Precalculus: Functions and Graphs

Question

The tallest man-made structure in the world is a television transmitting tower located near Mayville, North Dakota. From a distance of 1 mile on level ground, its angle of elevation is \(21^{\circ} 20^{\prime} 24^{\prime \prime} .\) Determine its height to the nearest foot.

Step-by-Step Solution

Answer

The tower's height is approximately 2068 feet.

Step 1: Convert Angle to Decimal Degrees

First, convert the angle of elevation from degrees, minutes, and seconds to a decimal degree. The angle given is \(21^{\circ} \, 20^{\prime} \, 24^{\prime \prime}\). There are 60 minutes in a degree and 60 seconds in a minute.\[\text{Decimal degrees} = 21 + \frac{20}{60} + \frac{24}{3600} \]\[\text{Decimal degrees} \approx 21.34^{\circ}\]

Step 2: Identify the Right Triangle Components

We have a right triangle where the base (adjacent side) is 1 mile, the angle of elevation is \(21.34^{\circ}\), and we need to find the height (opposite side) of the tower. We will use the tangent function, defined as:\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]

Step 3: Set Up the Tangent Equation

Using the tangent function, set up the equation to solve for the height.\[\tan(21.34^{\circ}) = \frac{h}{1 \text{ mile}}\]

Step 4: Calculate the Height

Rearrange the equation to solve for the height \(h\):\[h = 1 \times \tan(21.34^{\circ})\]Convert 1 mile to feet (1 mile = 5280 feet).\[h = 5280 \times \tan(21.34^{\circ})\]Calculate:\[h \approx 5280 \times 0.3918 \approx 2068 \text{ feet}\]

Step 5: Round the Height

Round the height calculated to the nearest foot.\[h \approx 2068 \text{ feet (to the nearest foot)}\]

Key Concepts

Angle of Elevation

The angle of elevation is a term used in trigonometry to describe the angle between the horizontal line from the observer's eye to the object being viewed and the line of sight. It is measured upwards from the horizontal. This concept is often used in real-life scenarios, such as determining the height of a structure or object from a particular point at ground level.
In our case, the angle of elevation is given as 21 degrees, 20 minutes, and 24 seconds, which are units to measure angles. When you look upwards at an angle towards the top of a tower from a flat, horizontal baseline, you measure the angle of elevation.

Tangent Function

The tangent function is one of the basic trigonometric functions used extensively in geometry and trigonometry. It relates the angles of a right triangle to the ratios of the lengths of its sides.
In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, it is expressed as:

• \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)

In our exercise, using the angle of elevation, we apply the tangent function to find the missing height of the structure. This function is convenient because it allows us to solve for an unknown side of the triangle if one side and the angle are known.

Right Triangle

A right triangle is a type of triangle that has one angle equal to 90 degrees. In real-world applications, right triangles are particularly useful for solving problems involving navigation, construction, and physics.
Our problem involves a right triangle where:

• One side represents the height of the tower (opposite side).
• Another side is the distance from the point of observation to the base of the tower (adjacent side).
• The right angle is formed where the ground meets this vertical height directly.

The hypotenuse is not needed here, as we're working specifically with the base and height using the tangent function. Understanding the parts of a right triangle is critical to setting up and solving trigonometry problems.

Degree Conversion

Degree conversion is a crucial step when working with angles. Angles are often given in degrees, minutes, and seconds, but can be easier to work with when converted to decimal degrees.
This process involves:

• 1 degree being equal to 60 minutes.
• 1 minute being equal to 60 seconds.

To convert an angle given in degrees, minutes, and seconds into a decimal degree, add the degrees to the minutes divided by 60, plus the seconds divided by 3600. For example, the conversion in our problem is:

• \[\text{Decimal degrees} = 21 + \frac{20}{60} + \frac{24}{3600} \approx 21.34^{\circ}\]

Using decimal degrees simplifies calculations and reduces errors when working through problems that involve trigonometric functions.