A Ladder Is Leaning Against The Wall Of A Building In Downtown Atlanta. When The Measure Of The Angle Between The Ladder And The Wall Is 12 Degrees, The Foot Of The Ladder Is 8 Ft Away From The Base Of The Wall. How Long Is The Ladder? Round To 1
Introduction
In this article, we will delve into a classic problem in trigonometry, involving a ladder leaning against a wall. The problem is as follows: a ladder is leaning against the wall of a building in downtown Atlanta. When the measure of the angle between the ladder and the wall is 12 degrees, the foot of the ladder is 8 ft away from the base of the wall. Our goal is to find the length of the ladder, rounded to the nearest foot.
Understanding the Problem
To approach this problem, we need to understand the concept of trigonometry and how it applies to right triangles. In a right triangle, the sine, cosine, and tangent functions relate the angles and side lengths of the triangle. In this case, we are given the angle between the ladder and the wall (12 degrees) and the distance from the foot of the ladder to the base of the wall (8 ft). We need to find the length of the ladder, which is the hypotenuse of the right triangle.
Using Trigonometry to Solve the Problem
We can use the cosine function to solve this problem. The cosine function is defined as the ratio of the adjacent side (the distance from the foot of the ladder to the base of the wall) to the hypotenuse (the length of the ladder). Mathematically, this can be expressed as:
cos(12°) = adjacent side / hypotenuse
We are given the adjacent side (8 ft) and the angle (12 degrees). We need to find the hypotenuse (the length of the ladder). Rearranging the equation, we get:
hypotenuse = adjacent side / cos(12°)
Calculating the Length of the Ladder
Now, we can plug in the values and calculate the length of the ladder. Using a calculator, we can find the value of cos(12°) and then divide the adjacent side (8 ft) by this value to get the length of the ladder.
cos(12°) ≈ 0.9781
hypotenuse ≈ 8 ft / 0.9781 hypotenuse ≈ 8.18 ft
Rounding to the nearest foot, the length of the ladder is approximately 8 ft.
Conclusion
In this article, we used trigonometry to solve a classic problem involving a ladder leaning against a wall. We applied the cosine function to find the length of the ladder, given the angle between the ladder and the wall and the distance from the foot of the ladder to the base of the wall. The length of the ladder was found to be approximately 8 ft, rounded to the nearest foot.
Additional Examples and Applications
This problem is a great example of how trigonometry can be applied to real-world situations. Here are a few additional examples and applications:
- Building design: Architects use trigonometry to design buildings and ensure that the structures are stable and secure.
- Surveying: Surveyors use trigonometry to measure distances and angles between landmarks and reference points.
- Physics: Trigonometry is used to describe the motion of objects in terms of position, velocity, and acceleration.
- Engineering: Engineers use trigonometry to design and optimize systems, such as bridges, tunnels, and other infrastructure projects.
Final Thoughts
In conclusion, this problem is a great example of how trigonometry can be used to solve real-world problems. By applying the cosine function, we were able to find the length of the ladder, given the angle between the ladder and the wall and the distance from the foot of the ladder to the base of the wall. This problem is a great starting point for exploring the many applications of trigonometry in mathematics and science.
References
- Trigonometry: A branch of mathematics that deals with the relationships between the sides and angles of triangles.
- Right triangle: A triangle with one angle equal to 90 degrees.
- Cosine function: A trigonometric function that relates the angle and side lengths of a right triangle.
- Hypotenuse: The longest side of a right triangle, opposite the right angle.
Glossary
- Adjacent side: The side of a right triangle that is next to the angle being measured.
- Angle: A measure of the amount of rotation between two lines or planes.
- Cosine: A trigonometric function that relates the angle and side lengths of a right triangle.
- Hypotenuse: The longest side of a right triangle, opposite the right angle.
- Trigonometry: A branch of mathematics that deals with the relationships between the sides and angles of triangles.
Introduction
In our previous article, we explored a classic problem in trigonometry involving a ladder leaning against a wall. We used the cosine function to find the length of the ladder, given the angle between the ladder and the wall and the distance from the foot of the ladder to the base of the wall. In this article, we will answer some common questions related to this problem and provide additional insights.
Q&A
Q: What is the formula for finding the length of the ladder?
A: The formula for finding the length of the ladder is:
hypotenuse = adjacent side / cos(12°)
Q: What is the value of cos(12°)?
A: The value of cos(12°) is approximately 0.9781.
Q: How do I calculate the length of the ladder using a calculator?
A: To calculate the length of the ladder using a calculator, follow these steps:
- Enter the value of the adjacent side (8 ft).
- Enter the value of the angle (12°).
- Press the "cos" button to calculate the cosine of the angle.
- Divide the adjacent side by the cosine of the angle to get the length of the ladder.
Q: What if I don't have a calculator? Can I still solve the problem?
A: Yes, you can still solve the problem without a calculator. You can use a trigonometric table or a calculator app on your phone to find the value of cos(12°).
Q: What is the significance of the angle between the ladder and the wall?
A: The angle between the ladder and the wall is an important factor in determining the length of the ladder. As the angle increases, the length of the ladder also increases.
Q: Can I use this formula to find the length of a ladder at any angle?
A: Yes, you can use this formula to find the length of a ladder at any angle. However, you will need to know the value of the cosine of the angle.
Q: What if the angle is greater than 90 degrees? Can I still use this formula?
A: No, you cannot use this formula if the angle is greater than 90 degrees. The formula is only valid for angles between 0 and 90 degrees.
Q: Can I use this formula to find the length of a ladder in a 3D space?
A: No, this formula is only valid for 2D spaces. If you need to find the length of a ladder in a 3D space, you will need to use a more complex formula that takes into account the x, y, and z coordinates of the ladder.
Conclusion
In this article, we answered some common questions related to the ladder problem and provided additional insights. We hope that this Q&A article has been helpful in clarifying any doubts you may have had about the problem.
Additional Resources
- Trigonometry: A branch of mathematics that deals with the relationships between the sides and angles of triangles.
- Right triangle: A triangle with one angle equal to 90 degrees.
- Cosine function: A trigonometric function that relates the angle and side lengths of a right triangle.
- Hypotenuse: The longest side of a right triangle, opposite the right angle.
Glossary
- Adjacent side: The side of a right triangle that is next to the angle being measured.
- Angle: A measure of the amount of rotation between two lines or planes.
- Cosine: A trigonometric function that relates the angle and side lengths of a right triangle.
- Hypotenuse: The longest side of a right triangle, opposite the right angle.
- Trigonometry: A branch of mathematics that deals with the relationships between the sides and angles of triangles.