Problem 8 - Precalculus

Question

From a point level with and 1000 ft away from the base of the Washington Monument, the angle of elevation to the top of the monument is \(29.05^{\circ} .\) Determine the height of the monument to the nearest half foot.

Step-by-Step Solution

Answer

The height of the Washington Monument is approximately 554.5 feet.

Step 1: Understand the Problem

We have a right triangle where the base is 1000 feet, the angle of elevation is \(29.05^{\circ}\), and the height of the monument is the opposite side to the angle. We need to find this height.

Step 2: Identify the Trigonometric Function

Since we know the angle of elevation and the adjacent side, and we need to find the opposite side, we use the tangent function. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side: \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).

Step 3: Set Up the Equation

Using the tangent function, we can set up the equation: \(\tan(29.05^{\circ}) = \frac{\text{height}}{1000}\).

Step 4: Solve for the Height

Rearrange the equation to solve for height: \(\text{height} = 1000 \cdot \tan(29.05^{\circ})\). Calculate this using a calculator: \(1000 \cdot \tan(29.05^{\circ}) \approx 554.23\text{ feet}\).

Step 5: Round the Answer

Round the calculated height to the nearest half foot: \(554.23\) feet rounds to approximately \(554.5\) feet.

Key Concepts

Angle of Elevation

In trigonometry, the angle of elevation is an important concept that helps us determine heights and distances. It refers to the angle formed by a horizontal line and the line of sight to an object above the horizontal. When standing a certain distance away from a tall object, like a building or monument, you look up to its top. The angle your line of sight makes with the horizontal is called the angle of elevation.
This angle is always measured from the horizontal line upwards to your line of sight. It's a key angle because it allows us to use trigonometric functions to find unknown distances or heights by creating a right triangle with known values. For example, in our exercise, the angle of elevation from a point 1000 feet away from the Washington Monument is given as \(29.05^{\circ}\). This angle, together with the distance from the base, helps us find the monument's height.

Tangent Function

The tangent function is one of the fundamental trigonometric functions used in right triangles. If you have an angle in a right triangle, the tangent of this angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, this is expressed as:

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

When you know the angle of elevation and the length of the base (adjacent side) of the right triangle, you can easily find the height or the opposite side using the tangent function.
As shown in our exercise, knowing the angle of elevation \(29.05^{\circ}\) and the distance from the base \(1000\) feet, we apply the tangent function to set up the equation \(\tan(29.05^{\circ}) = \frac{\text{height}}{1000}\). By rearranging and solving the equation, we find that the tangent function provides the necessary value to calculate the unknown height.

Right Triangle

A right triangle is a type of triangle that includes one right angle—an angle of exactly 90 degrees. This geometric shape is foundational in trigonometry because it allows the application of trigonometric ratios to find unknown sides or angles.
When you look at a right triangle in terms of its components, you have

• The hypotenuse, which is the longest side opposite the right angle.
• The opposite side, which is the side opposite the angle of interest.
• The adjacent side, which is the side next to the angle of interest and the right angle.

In the given exercise, the right triangle is formed by the horizontal distance from the Washington Monument (1000 feet), the height of the monument, and the line of sight to the top of the monument. The angle between the horizontal side (the adjacent side) and the line of sight (the hypotenuse) is the angle of elevation.”},{