In a right-angled triangle model of a ladder leaning against a vertical wall, the angle of elevation between the ladder and the horizontal ground is 60°. If the foot of the ladder is placed 4.6 m away from the wall along the ground, what is the full length (in metres) of the ladder?

Introduction:
This question is a classic application of trigonometry in height and distance problems. The situation, a ladder leaning against a wall, forms a right-angled triangle where the ladder is the hypotenuse. We are given the horizontal distance from the wall and the angle the ladder makes with the ground, and we must find the ladder's length.

Given Data / Assumptions:

• The ladder leans against a vertical wall.
• The angle between the ladder and the ground is 60°.
• The horizontal distance from the foot of the ladder to the wall is 4.6 m.
• The wall is perfectly vertical and the ground is horizontal, forming a right angle at the base.

Concept / Approach:
In a right-angled triangle, the cosine of an acute angle is defined as adjacent side divided by hypotenuse. Here:
cos(60°) = (horizontal distance) / (ladder length).We know cos(60°) = 1/2. So we can rearrange the formula to find the ladder length as:
ladder length = (horizontal distance) / cos(60°).

Step-by-Step Solution:
Step 1: Identify sides: adjacent side = 4.6 m, hypotenuse = ladder length (L).Step 2: Use the cosine relation: cos(60°) = 4.6 / L.Step 3: Substitute cos(60°) = 1/2, so 1/2 = 4.6 / L.Step 4: Rearrange to find L: L = 4.6 / (1/2) = 4.6 * 2.Step 5: Calculate: L = 9.2 m.

Verification / Alternative check:
If the ladder is 9.2 m long, then the horizontal projection should be 9.2 * cos(60°) = 9.2 * 0.5 = 4.6 m, which matches the given distance. This confirms that the calculation is consistent.

Why Other Options Are Wrong:
4.6 m would mean the ladder lies flat on the ground, not reaching the wall above ground level. Values like 2.3 m, 6.4 m, or 7.8 m are all shorter than the hypotenuse required by a 60° triangle with a 4.6 m base. None of them satisfy the cosine relation for 60° with the given adjacent side.

Common Pitfalls:
Students sometimes mistakenly use sine instead of cosine, or treat the 4.6 m as the hypotenuse. Remember: the ladder is the slant side (hypotenuse), the ground distance is adjacent to the angle with the ground, so cosine is the correct trigonometric ratio to use.

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