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.Which Measurements Are Accurate Based On The Scenario? Check All That Apply.- The Distance

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Introduction

In this scenario, a man is standing near the Washington Monument, observing the top of the monument at a $60^{\circ}$ angle of elevation from the ground. The height of the monument is given as 555 feet. We need to determine which measurements are accurate based on this scenario.

Understanding the Problem

To solve this problem, we need to use trigonometry, specifically the tangent function. The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

The Tangent Function

The tangent function can be represented mathematically as:

tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

In this scenario, the angle of elevation is $60^{\circ}$, and the height of the monument (opposite side) is 555 feet. We can use the tangent function to find the distance from the man to the monument (adjacent side).

Calculating the Distance

Using the tangent function, we can write:

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

To find the distance, we can rearrange the equation to solve for the distance:

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

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

tan⁡(60∘)≈1.732\tan(60^{\circ}) \approx 1.732

Now, we can plug this value into the equation to find the distance:

distance=5551.732≈320.5\text{distance} = \frac{555}{1.732} \approx 320.5

Accurate Measurements

Based on this calculation, we can conclude that the following measurements are accurate:

  • The height of the monument is 555 feet.
  • The angle of elevation is $60^{\circ}$.
  • The distance from the man to the monument is approximately 320.5 feet.

Conclusion

In this scenario, we used the tangent function to find the distance from the man to the monument. We can conclude that the height of the monument, the angle of elevation, and the distance from the man to the monument are all accurate measurements.

Discussion

This problem is a classic example of how trigonometry can be used to solve real-world problems. The tangent function is a powerful tool for finding the lengths of sides in right triangles. In this scenario, we used the tangent function to find the distance from the man to the monument, which is a critical piece of information for understanding the scenario.

Additional Considerations

It's worth noting that this problem assumes a right triangle, which is a fundamental concept in trigonometry. The tangent function is only defined for right triangles, so we must ensure that the scenario meets this condition. In this case, the angle of elevation is $60^{\circ}$, which is a right angle, so the scenario meets the condition.

Real-World Applications

This problem has real-world applications in fields such as architecture, engineering, and surveying. For example, architects and engineers use trigonometry to design buildings and structures, while surveyors use trigonometry to measure distances and angles in the field.

Conclusion

Introduction

In our previous article, we explored the scenario of a man standing near the Washington Monument, observing the top of the monument at a $60^{\circ}$ angle of elevation from the ground. We used the tangent function to find the distance from the man to the monument. In this article, we will answer some frequently asked questions related to this scenario.

Q&A

Q: What is the tangent function, and how is it used in this scenario?

A: The tangent function is a mathematical function that relates the angle of a right triangle to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In this scenario, we used the tangent function to find the distance from the man to the monument.

Q: Why is the angle of elevation $60^{\circ}$ important in this scenario?

A: The angle of elevation is $60^{\circ}$ because it is a right angle, which is a fundamental concept in trigonometry. The tangent function is only defined for right triangles, so we must ensure that the scenario meets this condition.

Q: How do we calculate the distance from the man to the monument?

A: To calculate the distance, we can use the tangent function:

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

We can rearrange the equation to solve for the distance:

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

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

tan⁡(60∘)≈1.732\tan(60^{\circ}) \approx 1.732

Now, we can plug this value into the equation to find the distance:

distance=5551.732≈320.5\text{distance} = \frac{555}{1.732} \approx 320.5

Q: What are the accurate measurements in this scenario?

A: The accurate measurements in this scenario are:

  • The height of the monument is 555 feet.
  • The angle of elevation is $60^{\circ}$.
  • The distance from the man to the monument is approximately 320.5 feet.

Q: What are some real-world applications of this scenario?

A: This scenario has real-world applications in fields such as architecture, engineering, and surveying. For example, architects and engineers use trigonometry to design buildings and structures, while surveyors use trigonometry to measure distances and angles in the field.

Q: Why is it important to understand the tangent function in this scenario?

A: Understanding the tangent function is crucial in this scenario because it allows us to find the distance from the man to the monument. The tangent function is a powerful tool for finding the lengths of sides in right triangles, and it is essential for solving problems like this one.

Conclusion

In conclusion, this Q&A article provides a comprehensive overview of the scenario of a man standing near the Washington Monument, observing the top of the monument at a $60^{\circ}$ angle of elevation from the ground. We used the tangent function to find the distance from the man to the monument, and we answered some frequently asked questions related to this scenario. We hope this article has provided valuable insights into the world of trigonometry and its applications.

Additional Resources

For more information on trigonometry and its applications, please refer to the following resources:

Conclusion

In conclusion, this Q&A article has provided a comprehensive overview of the scenario of a man standing near the Washington Monument, observing the top of the monument at a $60^{\circ}$ angle of elevation from the ground. We used the tangent function to find the distance from the man to the monument, and we answered some frequently asked questions related to this scenario. We hope this article has provided valuable insights into the world of trigonometry and its applications.