A Man Is Standing Near The Washington Monument. At A $60^{\circ}$ Angle Of Elevation From The Ground, The Man Sees The Top Of The 555-foot Monument.Check All That Apply For This Scenario:A. The Distance From The Man's Feet To The Base Of The

Introduction

In this scenario, we have a man standing near the Washington Monument, observing the top of the monument at a specific angle of elevation. The monument's height is given as 555 feet, and the angle of elevation is $60^{\circ}$. We need to determine the distance from the man's feet to the base of the monument. This problem involves trigonometry, specifically the use of right triangles and the sine, cosine, and tangent functions.

Understanding the Problem

To solve this problem, we need to visualize the situation and identify the relevant information. The man is standing at a certain distance from the base of the monument, and he observes the top of the monument at a $60^{\circ}$ angle of elevation. This forms a right triangle, with the distance from the man's feet to the base of the monument as the adjacent side, the height of the monument as the opposite side, and the line of sight from the man to the top of the monument as the hypotenuse.

Using Trigonometry to Solve the Problem

We can use the sine, cosine, and tangent functions to relate the angle of elevation to the sides of the right triangle. The sine function is defined as the ratio of the opposite side to the hypotenuse, the cosine function is defined as the ratio of the adjacent side to the hypotenuse, and the tangent function is defined as the ratio of the opposite side to the adjacent side.

In this case, we can use the sine function to relate the angle of elevation to the height of the monument and the distance from the man's feet to the base of the monument. The sine function is given by:

$$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$

where $\theta$ is the angle of elevation, the opposite side is the height of the monument (555 feet), and the hypotenuse is the line of sight from the man to the top of the monument.

Calculating the Distance

We can rearrange the sine function to solve for the distance from the man's feet to the base of the monument:

$$\text{distance} = \frac{\text{height}}{\tan(\theta)}$$

where $\theta$ is the angle of elevation ($60^{\circ}$) and the height is the height of the monument (555 feet).

Plugging in the Values

We can plug in the values into the equation to calculate the distance:

$$\text{distance} = \frac{555}{\tan(60^{\circ})}$$

Using a calculator to find the value of $\tan(60^{\circ})$, we get:

$$\tan(60^{\circ}) \approx 1.732$$

Now, we can plug this value into the equation:

$$\text{distance} = \frac{555}{1.732} \approx 320.5$$

Conclusion

Therefore, the distance from the man's feet to the base of the monument is approximately 320.5 feet.

Discussion

This problem involves the application of trigonometry to solve a real-world problem. The use of right triangles and the sine, cosine, and tangent functions allows us to relate the angle of elevation to the sides of the triangle and solve for the distance from the man's feet to the base of the monument.

Check all that apply for this scenario:

• A. The distance from the man's feet to the base of the monument is approximately 320.5 feet.
• B. The height of the monument is 555 feet.
• C. The angle of elevation is $60^{\circ}$.
• D. The line of sight from the man to the top of the monument is the hypotenuse of the right triangle.
• E. The sine function is used to relate the angle of elevation to the sides of the right triangle.
• F. The tangent function is used to relate the angle of elevation to the sides of the right triangle.
• G. The cosine function is used to relate the angle of elevation to the sides of the right triangle.

The correct answers are:

• A. The distance from the man's feet to the base of the monument is approximately 320.5 feet.
• B. The height of the monument is 555 feet.
• C. The angle of elevation is $60^{\circ}$.
• D. The line of sight from the man to the top of the monument is the hypotenuse of the right triangle.
• E. The sine function is used to relate the angle of elevation to the sides of the right triangle.
• F. The tangent function is used to relate the angle of elevation to the sides of the right triangle.
• G. The cosine function is used to relate the angle of elevation to the sides of the right triangle.
  

Q&A: Trigonometry and the Washington Monument

Q: What is the main concept used to solve this problem?

A: The main concept used to solve this problem is trigonometry, specifically the use of right triangles and the sine, cosine, and tangent functions.

Q: What is the angle of elevation in this scenario?

A: The angle of elevation is $60^{\circ}$.

Q: What is the height of the Washington Monument?

A: The height of the Washington Monument is 555 feet.

Q: What is the distance from the man's feet to the base of the monument?

A: The distance from the man's feet to the base of the monument is approximately 320.5 feet.

Q: What is the relationship between the angle of elevation and the sides of the right triangle?

A: The angle of elevation is related to the sides of the right triangle through the sine, cosine, and tangent functions.

Q: Which function is used to relate the angle of elevation to the sides of the right triangle?

A: The sine function is used to relate the angle of elevation to the sides of the right triangle.

Q: What is the formula used to calculate the distance from the man's feet to the base of the monument?

A: The formula used to calculate the distance from the man's feet to the base of the monument is:

$$\text{distance} = \frac{\text{height}}{\tan(\theta)}$$

where $\theta$ is the angle of elevation ($60^{\circ}$) and the height is the height of the monument (555 feet).

Q: What is the value of $\tan(60^{\circ})$?

A: The value of $\tan(60^{\circ})$ is approximately 1.732.

Q: How do you calculate the distance from the man's feet to the base of the monument using the formula?

A: To calculate the distance from the man's feet to the base of the monument using the formula, you need to plug in the values of the height and the angle of elevation into the formula and solve for the distance.

Q: What is the significance of the right triangle in this scenario?

A: The right triangle is significant in this scenario because it allows us to relate the angle of elevation to the sides of the triangle and solve for the distance from the man's feet to the base of the monument.

Q: What are the advantages of using trigonometry to solve this problem?

A: The advantages of using trigonometry to solve this problem include:

• It allows us to relate the angle of elevation to the sides of the right triangle.
• It provides a precise and accurate method for calculating the distance from the man's feet to the base of the monument.
• It is a powerful tool for solving problems involving right triangles and angles.

Q: What are the limitations of using trigonometry to solve this problem?

A: The limitations of using trigonometry to solve this problem include:

• It requires a good understanding of trigonometric concepts and formulas.
• It may not be suitable for solving problems involving complex or irregular shapes.
• It may not be applicable in situations where the angle of elevation is not known or is difficult to measure.

Q: How can you apply trigonometry to solve real-world problems?

A: You can apply trigonometry to solve real-world problems by:

• Identifying the relevant trigonometric concepts and formulas.
• Using right triangles and angles to relate the problem to the trigonometric concepts.
• Solving for the unknown quantities using the trigonometric formulas.
• Verifying the results using real-world data or measurements.

Q: What are some common applications of trigonometry in real-world problems?

A: Some common applications of trigonometry in real-world problems include:

• Navigation and surveying.
• Physics and engineering.
• Computer graphics and game development.
• Architecture and construction.
• Medical imaging and diagnostics.

Q: How can you improve your understanding of trigonometry and its applications?

A: You can improve your understanding of trigonometry and its applications by:

• Practicing problems and exercises.
• Studying real-world examples and case studies.
• Seeking help from teachers, tutors, or online resources.
• Joining study groups or online communities.
• Participating in math competitions or Olympiads.