Calculating the Length of a Ladder Against a Wall: A Comprehensive Guide

Determining the length of a ladder when it is placed against a wall at a specific angle and distance from the wall is a common problem in mathematics. This article provides a detailed explanation using trigonometry, specifically the cosine function, and offers multiple approaches to solve this problem accurately. Whether you are a student, a professional, or just curious about the solution, you will find this guide useful.

Introduction to the Problem

The problem we address involves a ladder leaning against a wall at an angle of 60°, with the base of the ladder 9 feet away from the wall. Our objective is to determine the length of the ladder using trigonometric principles.

Using Trigonometry to Solve the Problem

The cosine function in trigonometry helps us find the length of the hypotenuse in a right triangle when we know an angle and the length of the adjacent side. In this scenario, the angle is 60° and the adjacent side is 9 feet.

Step-by-Step Solution

Identify the given values:

Angle (theta) 60° Adjacent side 9 feet

Apply the cosine formula:

cos(theta)  adjacent / hypotenuse

Solve for the hypotenuse (length of the ladder):

cos(60°)  9 / length of the ladder

Substitute the known value of cos(60°) which is 0.5:

0.5  9 / length of the ladder

Isolate the length of the ladder:

length of the ladder  9 / 0.5

Calculate the length of the ladder:

length of the ladder  9 * 2  18 feet

Thus, the length of the ladder is 18 feet.

Alternative Methods for Verification

Let's explore alternative methods to verify the solution:

Using the Tangent Function

Identify the height from the ground to the top of the ladder:

h  9 ft * tan(60°)

Calculate the height:

h  9 ft * 1.732 (approx.)  15.6 ft (approx.)

Use the sine function to find the length of the ladder:

sin(60°)  h / length of the ladder

Solve for the length of the ladder:

sin(60°)  15.6 / length of the ladder

Substitute the known value of sin(60°) which is approximately 0.866:

0.866  15.6 / length of the ladder

Isolate the length of the ladder:

length of the ladder  15.6 / 0.866

Calculate the length of the ladder:

length of the ladder  18 ft

Thus, the length of the ladder is 18 feet.

Using the Pythagorean Theorem

Identify the sides of the triangle:

Height (h) 9 ft * tan(60°) 15.6 ft (approx.) Base 9 feet Hypotenuse (ladder length) unknown

Apply the Pythagorean Theorem:

h^2   base^2  ladder length^2

Substitute the known values:

(15.6)^2   9^2  ladder length^2

Calculate the squares:

243.36   81  ladder length^2

Add the squares:

324  ladder length^2

Take the square root:

ladder length  √324

Calculate the square root:

ladder length  18 feet

Thus, the length of the ladder is 18 feet.

Conclusion

Through the application of trigonometric functions, we have accurately determined the length of the ladder to be 18 feet. This problem highlights the importance of trigonometry in solving real-world problems. Whether you are using the cosine, tangent, or Pythagorean theorem, each method leads to the same conclusion.

Mixing these mathematical tools not only aids in problem-solving but also enhances your understanding of geometric principles. Whether you are a student, an engineer, or someone just interested in math, mastering these concepts can be a valuable skill.