Flagpole Height and Ladder Reach
During a school survey, two students stand on level ground on the same side of a vertical flagpole. From point A, the angle of elevation of the top of the pole is 30°. From point B, which is 20 m nearer the pole along the same straight line, the angle is 60°. The students’ instrument height is ignored because they sight from ground level. The school wants to replace a broken wire running from the top of the pole to a peg 6 m from the foot of the pole. A 20 m ladder is available for maintenance, and the safety rule says the ladder’s foot must be at least 6 m from the pole. A teacher suggests using 30° alone, but the students know the two observations must be combined. They also need to check whether the ladder can reach the top under the 6 m placement rule.
1. Find the height of the flagpole and the distance from B to the foot of the pole. Show the equations.
Read the worked answer
Let h be the pole height and x the distance from B to the foot. At B: tan 60°=h/x, so h=x√3. At A, the distance to the foot is x+20: tan 30°=h/(x+20), so h=(x+20)/√3.
Equate the two expressions for h: x√3=(x+20)/√3 3x=x+20 2x=20 x=10 m.
Then h=10√3≈17.32 m. So B is 10 m from the foot, and the flagpole is about 17.32 m high.
2. Find the length of wire needed from the top of the pole to the peg 6 m from the foot. Decide whether the 20 m ladder can reach the top when its foot is 6 m from the pole. State any assumption.
Read the worked answer
The wire, pole, and ground form a right triangle with legs h=10√3 m and 6 m. Wire length=√[(10√3)²+6²] =√(300+36)=√336=4√21≈18.33 m.
For the ladder placed 6 m from the foot, the required length to reach the top is the same √336≈18.33 m. Since 18.33 m<20 m, the 20 m ladder can reach the top with about 1.67 m to spare.
Assumption: the ground is level, the pole is vertical, and the ladder is straight and placed securely. The safety rule is satisfied because the foot is exactly 6 m from the pole.