A 16-foot Ladder Leans Against A Wall, Creating An Angle Of Elevation Of $55^{\circ}$ With The Ground. Find $x$, The Distance Between The Foot Of The Ladder And The Base Of The Building. Use Cosine To Solve For The Missing Side

by ADMIN 232 views

Introduction

When a ladder leans against a wall, it creates an angle of elevation with the ground. This angle is a crucial piece of information in solving for the distance between the foot of the ladder and the base of the building. In this article, we will use the cosine function to find the missing side of the right triangle formed by the ladder, the wall, and the ground.

Understanding the Problem

We are given a 16-foot ladder that leans against a wall, creating an angle of elevation of $55^{\circ}$ with the ground. Our goal is to find the distance between the foot of the ladder and the base of the building, denoted as $x$. To solve for $x$, we will use the cosine function, which relates the angle, the adjacent side, and the hypotenuse of a right triangle.

Using Cosine to Solve for the Missing Side

The cosine function is defined as:

cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}

In our problem, the angle $\theta$ is $55^{\circ}$, the adjacent side is $x$, and the hypotenuse is 16 feet (the length of the ladder). Plugging these values into the cosine function, we get:

cos⁡(55∘)=x16\cos(55^{\circ}) = \frac{x}{16}

To solve for $x$, we can multiply both sides of the equation by 16:

x=16cos⁡(55∘)x = 16 \cos(55^{\circ})

Calculating the Value of $x$

Now that we have the equation for $x$, we can calculate its value using a calculator. Plugging in the value of $\cos(55^{\circ})$, we get:

x=16cos⁡(55∘)≈16×0.5736≈9.18x = 16 \cos(55^{\circ}) \approx 16 \times 0.5736 \approx 9.18

Therefore, the distance between the foot of the ladder and the base of the building is approximately 9.18 feet.

Conclusion

In this article, we used the cosine function to solve for the missing side of a right triangle formed by a 16-foot ladder, a wall, and the ground. We found that the distance between the foot of the ladder and the base of the building is approximately 9.18 feet. This problem demonstrates the importance of trigonometry in solving real-world problems involving right triangles.

Real-World Applications

The problem of finding the distance between the foot of a ladder and the base of a building has many real-world applications. For example:

  • Construction: In construction, it is essential to determine the distance between the foot of a ladder and the base of a building to ensure that the ladder is placed safely and securely.
  • Surveying: In surveying, it is crucial to determine the distance between the foot of a ladder and the base of a building to establish the boundaries of a property.
  • Architecture: In architecture, it is essential to determine the distance between the foot of a ladder and the base of a building to design and construct buildings that are safe and functional.

Tips and Tricks

Here are some tips and tricks to help you solve problems involving right triangles and trigonometry:

  • Use the correct trigonometric function: Make sure to use the correct trigonometric function (sine, cosine, or tangent) to solve the problem.
  • Identify the angle and the sides: Identify the angle and the sides of the right triangle to determine which trigonometric function to use.
  • Use a calculator: Use a calculator to calculate the values of trigonometric functions and solve for the missing side.
  • Check your work: Check your work by plugging in the values of the sides and the angle to ensure that the solution is correct.

Common Mistakes

Here are some common mistakes to avoid when solving problems involving right triangles and trigonometry:

  • Using the wrong trigonometric function: Using the wrong trigonometric function can lead to incorrect solutions.
  • Not identifying the angle and the sides: Not identifying the angle and the sides can lead to incorrect solutions.
  • Not using a calculator: Not using a calculator can lead to incorrect solutions.
  • Not checking your work: Not checking your work can lead to incorrect solutions.

Conclusion

In conclusion, the problem of finding the distance between the foot of a ladder and the base of a building is a classic example of a right triangle problem that can be solved using trigonometry. By using the cosine function, we can find the missing side of the right triangle and determine the distance between the foot of the ladder and the base of the building. This problem demonstrates the importance of trigonometry in solving real-world problems involving right triangles.

Introduction

In our previous article, we used the cosine function to solve for the missing side of a right triangle formed by a 16-foot ladder, a wall, and the ground. We found that the distance between the foot of the ladder and the base of the building is approximately 9.18 feet. In this article, we will answer some frequently asked questions about the problem and provide additional information to help you understand the concept better.

Q&A

Q: What is the angle of elevation in the problem?

A: The angle of elevation in the problem is 55 degrees.

Q: What is the length of the ladder?

A: The length of the ladder is 16 feet.

Q: What is the distance between the foot of the ladder and the base of the building?

A: The distance between the foot of the ladder and the base of the building is approximately 9.18 feet.

Q: How do I use the cosine function to solve for the missing side?

A: To use the cosine function to solve for the missing side, you need to identify the angle, the adjacent side, and the hypotenuse of the right triangle. Then, you can plug these values into the cosine function and solve for the missing side.

Q: What is the formula for the cosine function?

A: The formula for the cosine function is:

cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}

Q: How do I calculate the value of the missing side?

A: To calculate the value of the missing side, you need to multiply the length of the hypotenuse by the cosine of the angle.

Q: What are some real-world applications of the problem?

A: Some real-world applications of the problem include construction, surveying, and architecture.

Q: What are some common mistakes to avoid when solving problems involving right triangles and trigonometry?

A: Some common mistakes to avoid when solving problems involving right triangles and trigonometry include using the wrong trigonometric function, not identifying the angle and the sides, not using a calculator, and not checking your work.

Additional Tips and Tricks

Here are some additional tips and tricks to help you solve problems involving right triangles and trigonometry:

  • Use a calculator: Use a calculator to calculate the values of trigonometric functions and solve for the missing side.
  • Check your work: Check your work by plugging in the values of the sides and the angle to ensure that the solution is correct.
  • Use a diagram: Use a diagram to visualize the problem and identify the angle, the adjacent side, and the hypotenuse.
  • Practice, practice, practice: Practice solving problems involving right triangles and trigonometry to become more comfortable with the concepts and formulas.

Conclusion

In conclusion, the problem of finding the distance between the foot of a ladder and the base of a building is a classic example of a right triangle problem that can be solved using trigonometry. By using the cosine function, we can find the missing side of the right triangle and determine the distance between the foot of the ladder and the base of the building. We hope that this Q&A article has provided you with a better understanding of the concept and has helped you to solve similar problems in the future.

Common Misconceptions

Here are some common misconceptions about the problem:

  • The angle of elevation is always 90 degrees: The angle of elevation is not always 90 degrees. It can be any angle between 0 and 90 degrees.
  • The length of the ladder is always the hypotenuse: The length of the ladder is not always the hypotenuse. It can be either the hypotenuse or one of the legs of the right triangle.
  • The distance between the foot of the ladder and the base of the building is always the same: The distance between the foot of the ladder and the base of the building is not always the same. It depends on the angle of elevation and the length of the ladder.

Real-World Examples

Here are some real-world examples of the problem:

  • Construction: In construction, it is essential to determine the distance between the foot of a ladder and the base of a building to ensure that the ladder is placed safely and securely.
  • Surveying: In surveying, it is crucial to determine the distance between the foot of a ladder and the base of a building to establish the boundaries of a property.
  • Architecture: In architecture, it is essential to determine the distance between the foot of a ladder and the base of a building to design and construct buildings that are safe and functional.

Conclusion

In conclusion, the problem of finding the distance between the foot of a ladder and the base of a building is a classic example of a right triangle problem that can be solved using trigonometry. By using the cosine function, we can find the missing side of the right triangle and determine the distance between the foot of the ladder and the base of the building. We hope that this Q&A article has provided you with a better understanding of the concept and has helped you to solve similar problems in the future.