Question

10 from an observer o who is 200 m from a building, the angles of elevation to the bottom and top of a flagpole are ( 36^{circ} ) and ( 38^{circ} ) respectively. find the height of the flagpole.

Explanation:

Step1: Calculate the height from the ground to the bottom of the flagpole

Let \(h_1\) be the height from the ground to the bottom of the flagpole. Using the tangent function \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), for \(\theta = 36^{\circ}\) and adjacent side \(= 200\)m.
\(\tan36^{\circ}=\frac{h_1}{200}\), so \(h_1 = 200\times\tan36^{\circ}\)
\(\tan36^{\circ}\approx0.7265\), then \(h_1\approx200\times0.7265 = 145.3\)m

Step2: Calculate the height from the ground to the top of the flagpole

Let \(h_2\) be the height from the ground to the top of the flagpole. For \(\theta = 38^{\circ}\) and adjacent side \(= 200\)m.
\(\tan38^{\circ}=\frac{h_2}{200}\), so \(h_2=200\times\tan38^{\circ}\)
\(\tan38^{\circ}\approx0.7813\), then \(h_2\approx200\times0.7813 = 156.26\)m

Step3: Calculate the height of the flagpole

Let \(h\) be the height of the flagpole. \(h=h_2 - h_1\)
\(h = 200\times(\tan38^{\circ}-\tan36^{\circ})\)
Substitute the values of \(\tan38^{\circ}\) and \(\tan36^{\circ}\)
\(h\approx200\times(0.7813 - 0.7265)=200\times0.0548 = 10.96\)m

Answer:

The height of the flagpole is approximately \(11.0\)m.