Question

Two vertical poles are 40 metres apart and the height of one is double that of the other. From the middle point of the line joining their feet, an observer finds the angular elevations of their tops to be complementary. Find their heights.

• A. 14.14 m, 28.28 m
• B. 12 m, 22 m
• C. 12.12 m, 24.24 m
• D. None of these.

Answer 1

The Correct Option is A

Given: Two vertical poles are 40 metres apart. The height of one pole is double that of the other. From the midpoint of the line joining their feet, the angular elevations of their tops are complementary.

Let the height of the shorter pole be \(h\) meters. Then the height of the taller pole is \(2h\) meters.

The distance from the midpoint of the line joining their feet to each pole is \( \frac{40}{2} = 20 \) meters.

Let \(\theta\) be the angle of elevation to the top of the shorter pole, and \(90^\circ - \theta\) be the angle of elevation to the top of the taller pole (since they are complementary).

For the shorter pole:

\(\tan \theta = \frac{h}{20}\)

For the taller pole:

\(\tan(90^\circ - \theta) = \frac{2h}{20}\)

Since \(\tan(90^\circ - \theta) = \cot \theta\), we have:

\(\cot \theta = \frac{2h}{20}\)

We know that \(\tan \theta \cdot \cot \theta = 1\), thus:

\(\frac{h}{20} \cdot \frac{2h}{20} = 1\)

\(\frac{2h^2}{400} = 1\)

\(2h^2 = 400\)

\(h^2 = 200\)

\(h = \sqrt{200} = 10\sqrt{2} \approx 14.14\) meters.

Thus, the height of the taller pole is \(2h = 2 \times 14.14 = 28.28\) meters.

Shorter pole height	14.14 m
Taller pole height	28.28 m