In a right-triangle situation, a ladder is leaning against a perfectly vertical wall and makes an angle of elevation of 60° with the horizontal ground. If the foot of the ladder is placed 4.6 m away from the wall, what is the length of the ladder in metres?

Introduction / Context:
This height and distance question applies basic trigonometry to a real-life scenario where a ladder leans against a wall. You are expected to use the angle of elevation and the horizontal distance from the wall to determine the actual length of the ladder, which forms the hypotenuse of a right triangle.

Given Data / Assumptions:

• The wall is perfectly vertical and the ground is perfectly horizontal, so the ladder, wall and ground form a right triangle.
• Angle of elevation between the ladder and the ground = 60°.
• Horizontal distance between the foot of the ladder and the wall = 4.6 m.
• We must find the length of the ladder in metres.

Concept / Approach:
When a ladder leans against a wall, the ladder is the hypotenuse of a right triangle, the distance from the wall is the adjacent side, and the height of contact with the wall is the opposite side. The cosine of the angle of elevation is defined as adjacent / hypotenuse. Therefore, we can use the cosine formula:
cos(theta) = adjacent / hypotenuseand rearrange it to solve for the hypotenuse (the ladder length).

Step-by-Step Solution:
Let L be the length of the ladder.Angle of elevation, theta = 60°.Adjacent side (distance from wall) = 4.6 m.cos(60°) = adjacent / hypotenuse = 4.6 / LWe know cos(60°) = 1 / 2.So, 1 / 2 = 4.6 / LCross-multiplying gives: L = 4.6 * 2 = 9.2Therefore, the ladder length L = 9.2 m.

Verification / Alternative check:
We can quickly check using the sine relation. The vertical height h would be:
h = L * sin(60°) = 9.2 * (√3 / 2)This gives a positive realistic height value, confirming that the calculated hypotenuse is reasonable for the given horizontal distance of 4.6 m and angle 60°.

Why Other Options Are Wrong:
7.8 m: Too short; with a 60° angle and base 4.6 m it would imply cos(60°) ≠ 4.6 / 7.8.4.6 m: This would correspond to a 60° angle only if the ladder were equal to the base, which contradicts the cosine relationship.2.3 m: Much too short; the ladder could not even reach the wall at any realistic angle.8.4 m: Closer, but still inconsistent with cos(60°) = 4.6 / L.

Common Pitfalls:
Students often confuse sine and cosine and might incorrectly use sin(60°) = opposite / hypotenuse with the base instead of the height. Another common mistake is to assume the ladder length equals the base distance. Also, forgetting that cos(60°) = 1 / 2 can lead to incorrect numerical values.

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