A ladder leans against a vertical wall making an angle of 60° with the ground. If the foot of the ladder is 4.6 m away from the wall, what is the length of the ladder?

Introduction / Context:
This is a classic right triangle problem involving a ladder leaning against a vertical wall. The ladder, wall and ground form a right angled triangle where the ladder is the hypotenuse. You are given the angle between the ladder and the ground, as well as the distance from the foot of the ladder to the wall. The question asks for the length of the ladder itself.

Given Data / Assumptions:

• The ladder makes an angle of 60 degrees with the horizontal ground.
• The foot of the ladder is 4.6 m away from the wall.
• The wall is vertical and the ground is horizontal, forming a right angle at the foot of the wall.
• The ladder forms the hypotenuse of the right triangle.
• Standard value: cos 60 = 1 / 2.

Concept / Approach:
In a right angled triangle, the cosine of the angle at the base is given by adjacent / hypotenuse, where the adjacent side is along the ground and the hypotenuse is the ladder. Here, we know the adjacent side (4.6 m) and the angle (60 degrees) and need to find the hypotenuse (ladder length). So, hypotenuse = adjacent / cos θ. Using cos 60 = 1 / 2, we can directly compute the length.

Step-by-Step Solution:
Let L be the length of the ladder. The angle between the ladder and the ground is 60 degrees. By definition, cos 60 = adjacent / hypotenuse = 4.6 / L. Since cos 60 = 1 / 2, we have 1 / 2 = 4.6 / L. Cross multiplying gives L = 4.6 × 2 = 9.2 m.

Verification / Alternative check:
If L = 9.2 m and the base is 4.6 m, then the height of the ladder on the wall is sqrt(L² - base²) = sqrt(9.2² - 4.6²). This simplifies to sqrt(84.64 - 21.16) = sqrt(63.48) which is about 7.97 m. The sine of 60 degrees is about 0.866, and 0.866 × 9.2 ≈ 7.97 m. This matches the computed height, confirming that the triangle satisfies both sine and cosine relationships for 60 degrees.

Why Other Options Are Wrong:

• 2.3 m: This is only half of the base distance and far too short for a ladder reaching up a wall.
• 4.6 m: This equals the base distance and would correspond to a 45 degree angle, not 60 degrees.
• 7.8 m and 8.0 m: These are less than the true hypotenuse computed using cos 60 and do not satisfy the exact trigonometric ratio for 60 degrees.

Common Pitfalls:
A common mistake is to use sine instead of cosine with the base or to misidentify which side is adjacent and which is opposite. Another error is to compute L = base × cos 60 instead of base / cos 60, which would incorrectly shorten the ladder. Always remember that hypotenuse is longer than either leg and check if your answer is logically larger than the given base distance.

Final Answer:
The length of the ladder is 9.2 m.