Leaning ladder against a wall: A ladder makes a 60° angle with the horizontal ground, and its foot is 4.6 m away from the wall. Find the length of the ladder.
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A2.3 m
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B4.6 m
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C7.8 m
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D9.2 m
Answer
Correct Answer: 9.2 m
Explanation
Introduction / Context:A ladder against a wall forms a right triangle with the ground (adjacent) and wall (opposite). The angle with ground and the horizontal distance to the wall are known; find the hypotenuse (ladder length).
Given Data / Assumptions:
- Angle with ground = 60°.
- Adjacency (foot to wall) = 4.6 m.
- Ladder is straight; wall and ground are perpendicular.
Concept / Approach:Use cos θ = adjacent / hypotenuse → hypotenuse = adjacent / cos θ.
Step-by-Step Solution:
cos 60° = 1/2.Length L = 4.6 / (1/2) = 9.2 m.Verification / Alternative check:Compute vertical reach: L * sin 60° = 9.2 * (√3/2) ≈ 7.97 m; Pythagoras with 4.6 m confirms hypotenuse ≈ 9.2 m.
Why Other Options Are Wrong:2.3 and 4.6 m are too short; 7.8 m corresponds to cos near 0.59, not 0.5.
Common Pitfalls:Using tan instead of cos; mixing degrees with radians; rounding too early on √3 or cos values.
Final Answer:9.2 m