Flagpole Height and Rope Length
During a school survey, a Class X team is measuring the height of the flagpole in the school ground. The students stand at two points P and Q on the same straight line from the foot of the pole, with Q farther away. From P, the angle of elevation of the top of the pole is 60°. From Q, 20 m farther from the pole, the angle is 30°. The students' clinometer is held at a height of 1.5 m above the ground at both points. The team leader claims the flagpole is 18 m high and wants to order an 18 m rope to reach from the ground at the foot of the pole to the top. The teacher asks the team to check the claim, decide whether the rope is long enough, and state how much extra rope would be needed if the calculated height is rounded up to the next whole metre.
1. Using the two angles of elevation and the 20 m distance, calculate the actual height of the flagpole.
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Let h be the height of the pole above the clinometer and x the horizontal distance from P to the foot. From P: tan60 = h/x, so x = h/sqrt3. From Q: tan30 = h/(x+20), so x+20 = h sqrt3. Substitute: h/sqrt3 + 20 = h sqrt3. 20 = h(sqrt3 − 1/sqrt3) = h(2/sqrt3). h = 10 sqrt3 ≈ 17.32 m. Actual pole height = h + 1.5 = 18.82 m.
2. Check the leader's 18 m claim and decide whether an 18 m rope from the ground at the foot will reach the top.
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The leader's 18 m claim is too low by about 0.82 m. An 18 m rope from the foot to the top would not reach; it is short by about 0.82 m. The correct height is about 18.82 m.
3. If the calculated height is rounded up to the next whole metre, what length should be ordered and how much extra is that over the proposed 18 m rope?
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Rounded up to the next whole metre, the height is 19 m. Compared with the 18 m rope, the extra length needed is 19 − 18 = 1 m. Compared with the calculated 18.82 m, the extra is 19 − 18.82 = 0.18 m, but for ordering a whole-metre rope the required extra over 18 m is 1 m.