Problem 308 - Precalculus

Question

The angle of elevation to the top of a building in New York is found to be 9 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.

Step-by-Step Solution

Answer

The building is approximately 836.35 feet tall.

Step 1: Convert the Distance to Feet

The problem states the distance from the observer to the building is 1 mile. To work with this measurement, we need to convert miles to feet. There are 5280 feet in a mile. Therefore, the distance is 1 mile = 5280 feet.

Step 2: Understand the Trigonometric Relationship

The angle of elevation and the distance to the building form a right triangle, with the building's height as the opposite side, the distance to the building as the adjacent side, and the angle of elevation as the given angle. We will use the tangent function in trigonometry:\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\] where \(\theta\) is the angle of elevation.

Step 3: Use Tangent to Find the Height

Substitute the known values into the tangent formula. We know \(\theta = 9^{\circ}\) and the adjacent side is 5280 feet. The formula becomes:\[\tan(9^{\circ}) = \frac{h}{5280}\]where \(h\) is the height of the building. Solve for \(h\) by multiplying both sides by 5280:\[h = 5280 \times \tan(9^{\circ})\].

Step 4: Calculate the Height

Use a calculator to compute \(\tan(9^{\circ})\) and then calculate the height of the building:\[h \approx 5280 \times 0.1584 \approx 836.35\text{ feet}\]. So, the height of the building is approximately 836.35 feet.

Key Concepts

Angle of Elevation

The angle of elevation is a commonly used trigonometric concept when measuring heights without direct measurement. Imagine you are standing a certain distance away from a tall building or another object. If you look straight ahead, your line of sight is horizontal. When you tilt your head up to view the top of the building, the angle formed between your sight line and the horizontal is the angle of elevation.

To visualize this, think of a triangle where:

• One leg is the distance from you to the base of the building (the adjacent side).
• The other leg is the vertical line from the top of the building to the same horizontal level as your eyes (the opposite side).
• The angle of elevation forms the acute angle at your position, creating a right triangle.

This angle is crucial as it helps us use trigonometric functions, like the tangent, to calculate unknown distances or heights.

Tangent Function

The tangent function is one of the primary trigonometric functions. It relates the angles of a right triangle to the proportions of its sides. Specifically, for any given angle \(\theta\) in a right triangle, the tangent function is defined as the ratio of the length of the opposite side to the length of the adjacent side. This relationship is expressed mathematically as:

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

In the context of our problem, the angle of elevation serves as \(\theta\), the height of the building is the opposite side, and the distance from the observer to the building (5280 feet, after conversion) is the adjacent side. By knowing any two of these three quantities, you can always calculate the third. Hence, when we know the angle and the distance to the building, we can rearrange the tangent formula to find the height of the building.

Right Triangle

A right triangle is a type of triangle that has one angle equal to 90 degrees. This makes it highly useful in trigonometry, as the relationships between angles and sides become straightforward.

• The side opposite the right angle is called the hypotenuse, which is the longest side.
• The other two sides, the opposite and adjacent, form the right angle.
• Each of these sides interacts with an angle in a way described by trigonometric functions like sine, cosine, and tangent.

In our exercise, the problem sets up a right triangle where:

• The hypotenuse is not directly involved.
• The opposite side represents the height of the building we seek to find.
• The adjacent side is the distance from the observer to the base, which is 5280 feet.
• The angle is given as 9 degrees.

With one angle and one side known, the properties of right triangles allow us to use trigonometry to solve for the remaining sides.

Distance Conversion

Distance conversion is an essential mathematical skill, particularly in real-world applications when measurements are given in different units. In our exercise, the distance from the observer to the building is initially provided in miles, but to solve the problem using trigonometric functions, it's often necessary to convert that to feet.

Here's how distance conversion generally works:

• It relies on knowing the conversion factors between units.
• For instance, one mile equals 5280 feet.
• To convert miles to feet, multiply the number of miles by this conversion factor.

In our problem:

The given distance is 1 mile, and applying the conversion gives us: \(1 \text{ mile} \times 5280 \frac{\text{feet}}{\text{mile}} = 5280 \text{ feet}\).

Such conversions ensure that all units are consistent, which is crucial when utilizing formulas and functions that require specific units like the tangent function in trigonometry.