Flagpole Height from Opposite Angles of Elevation
Two students measure the height of a flagpole from opposite sides of its base on level ground. Their eye level is 1.4 m above the ground. From the nearer position, the angle of elevation to the top of the pole is 60°; from the farther position, directly opposite, the angle is 30°. The horizontal distance between the two students is 20 m. The school brochure claims the flagpole is 12 m tall. A teacher warns that the students must add their eye height and must not simply add the two horizontal distances to the pole. The students also plan to check whether a 10 m rope tied at the top and anchored at the nearer observation point would reach the ground without slack.
1. Calculate the height of the flagpole above eye level and the total height of the flagpole. Compare the total height with the brochure's 12 m claim.
Read the worked answer
Let h be the height of the pole above eye level. At the nearer point the horizontal distance to the base is h/tan 60° = h/√3. At the farther point it is h/tan 30° = h√3.
Because the students are on opposite sides, these distances add to 20 m: h/√3 + h√3 = 20 h(1/√3 + √3) = 20 h(4/√3) = 20 h = 5√3 ≈ 8.66 m.
Total pole height = 8.66 + 1.4 = 10.06 m, approximately 10.1 m. The brochure's 12 m overstates the height by about 1.94 m, so the claim is not supported.
2. Determine whether a 10 m rope tied at the top and anchored at the nearer observation point would reach without slack. Use the nearer horizontal distance and the total pole height.
Read the worked answer
The nearer horizontal distance is h/√3 = 5√3/√3 = 5 m. The vertical drop from the top to the ground at that point is the full pole height, about 10.06 m.
Rope needed = √(5² + 10.06²) ≈ √(25 + 101.2) = √126.2 ≈ 11.23 m.
A 10 m rope is too short by about 1.2 m; it would not reach the ground without slack if tied at the top and anchored at that point. Using only h = 8.66 m and forgetting the 1.4 m eye height would understate the rope length.
3. The 20 m distance is measured to the nearest metre. Find a defensible range for the total flagpole height and state whether the 12 m brochure claim survives that rounding.
Read the worked answer
Since 20 m is rounded to the nearest metre, the true distance D satisfies 19.5 ≤ D < 20.5 m.
From h = D × √3/4:
- Lower h ≈ 19.5 × 0.4330 = 8.44 m, total ≈ 9.84 m.
- Upper h ≈ 20.5 × 0.4330 = 8.88 m, total ≈ 10.28 m.
A defensible range is about 9.8 m to 10.3 m. Even at the top of the range, the pole is well below 12 m, so the brochure's claim does not survive the measurement uncertainty.