Flagpole Height and Overhead Cable Clearance
During a safety inspection, two students measure the angle of elevation of the top of the school flagpole from two points on level ground in line with its base. From the nearer point the angle is 60°; from the farther point, 20 m away, the angle is 30°. Their clinometer is held at eye level, 1.5 m above the ground. A horizontal overhead cable runs above the flagpole at a height of 20 m. The school plans to replace the flagpole with a 19.5 m mast, and the safety rule requires at least 0.5 m clearance between the top of the mast and the cable. The students first forgot to add their eye height and got a height of about 17.3 m; the caretaker says that is close enough. The head teacher wants the actual height and a clear decision on whether the replacement mast is allowed.
1. Using the two angles, derive the height of the flagpole above the students' eye level.
Read the worked answer
Let $h$ be the height of the flagpole above eye level, and let $d$ be the horizontal distance from the nearer point to the base.
At the nearer point,
$$\tan60^\circ=\frac{h}{d} \Rightarrow d=\frac{h}{\sqrt3}.$$
At the farther point, the distance is $d+20$, so
$$\tan30^\circ=\frac{h}{d+20} \Rightarrow d+20=h\sqrt3.$$
Substitute $d=\frac{h}{\sqrt3}$:
$$\frac{h}{\sqrt3}+20=h\sqrt3.$$
Multiply by $\sqrt3$:
$$h+20\sqrt3=3h \Rightarrow 2h=20\sqrt3 \Rightarrow h=10\sqrt3\approx17.32\text{ m}.$$
So the top is about 17.32 m above eye level.
2. Find the actual height of the flagpole and check its clearance below the 20 m cable.
Read the worked answer
The clinometer was held 1.5 m above ground, so the actual height is
$$H=10\sqrt3+1.5\approx17.32+1.5=18.82\text{ m}.$$
Clearance below the cable is
$$20-18.82=1.18\text{ m}.$$
This is greater than the required 0.5 m, so the existing flagpole is acceptable. The students' earlier value of 17.3 m ignored the 1.5 m eye height and understated the height; the caretaker's 'close enough' comment is not acceptable for a safety decision.
3. Decide whether the planned 19.5 m replacement mast satisfies the clearance rule.
Read the worked answer
With a 19.5 m mast, the clearance is
$$20-19.5=0.5\text{ m}.$$
The rule requires at least 0.5 m, so the mast is exactly at the limit. In practice, this leaves no tolerance for measurement or construction error. If the mast is exactly 19.5 m and the cable is exactly 20 m above the same ground level, it is allowed; otherwise it could violate the rule. The safer decision is to use a mast no taller than 19.5 m and verify the cable height on site before installation.