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Chapter 7: Problem 80

Washington Monument The angle of elevation of the Sun is \(35.1^{\circ}\) at the instant the shadow cast by the Washington Monument is 789 feet long. Use this information to calculate the height of the monument.

Short Answer

Expert verified
The height of the Washington Monument is approximately 556 feet.

Step by step solution

01

- Identify the right triangle

In this problem, the Washington Monument, its shadow, and the line from the top of the monument to the tip of its shadow form a right-angled triangle. The height of the monument is the side opposite the angle of elevation, and the length of the shadow is the side adjacent to the angle.
02

- Use the tangent function

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, \(\tan(\theta) = \frac{opposite}{adjacent}\). Here, \(\theta = 35.1^{\text{circ}}\), the 'opposite' side is the height of the monument, and the 'adjacent' side is the length of the shadow (789 feet).
03

- Set up the equation

Using the tangent function: \(\tan(35.1^{\text{circ}}) = \frac{height}{789 \text{ feet}}\).
04

- Solve for the height

Rearrange the equation to solve for the height: \(height = 789 \times \tan(35.1^{\text{circ}})\). Next, calculate the tangent: \(\tan(35.1^{\text{circ}}) \approx 0.7045\). Finally, multiply: \(height \approx 789 \times 0.7045 \approx 556\).
05

- Final answer

The height of the Washington Monument is approximately 556 feet.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angle of Elevation
The angle of elevation is an important concept in trigonometry, especially when solving problems involving right triangles. It refers to the angle between the line of sight when looking up at an object and the horizontal plane. Imagine standing some distance away from a tall structure like the Washington Monument and looking up at its peak. The angle your line of sight makes with the ground is the angle of elevation.

These angles are crucial in real-life applications such as navigation, architecture, and even in simple tasks like measuring the height of an object. By knowing the angle and one side of a right triangle, one can calculate other dimensions using trigonometric functions.
  • In this example, the angle of elevation of the Sun is given as 35.1掳.
  • The line of sight forms an angle of 35.1掳 with the horizontal ground.
Understanding this angle helps us set up our trigonometric functions correctly to find unknown sides of a right triangle.
Right Triangle
In the given problem, we are dealing with a right triangle. Right triangles are triangles where one angle measures exactly 90掳. Because of this specific angle, right triangles have unique properties and relationships defined by trigonometry.

Here鈥檚 how the right triangle is formed in our scenario:
  • The height of the Washington Monument forms the vertical side (opposite side) of the right triangle.
  • The shadow of the monument forms the horizontal side (adjacent side) of the right triangle.
  • The hypotenuse is the line segment extending from the top of the monument to the tip of its shadow.
The positioning of these sides makes it possible to apply basic trigonometric ratios like sine, cosine, and tangent to figure out unknown lengths based on known values. In this context, we use the tangent function because it relates the height and the length of the shadow.
Tangent Function
The tangent function is key when dealing with right triangles and angles of elevation. It connects the angle with the lengths of the opposite and adjacent sides. The tangent of an angle \theta is given by the formula:

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

In our problem, \theta is the angle of elevation (35.1掳). We are looking to find the height (opposite side) of the Washington Monument using the known length of its shadow (adjacent side). Following these steps:
  • Identify the given values: \theta = 35.1掳, adjacent side = 789 feet.
  • Set up the equation: \[\tan{35.1^{\circ}} = \frac{height}{789}\].
  • Rearrange the equation to solve for height: \[height = 789 \times \tan{35.1^{\circ}}\].
  • Find \tan{35.1^{\circ}}, which is approximately 0.7045.
  • Calculate the height: height 鈮� 789 脳 0.7045 鈮� 556 feet.
By understanding and applying the tangent function, we accurately determine that the height of the Washington Monument is approximately 556 feet.

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