Question

surveyor stands 90m from the base of a tower on which an antenna stands. He measures the angles of elevation to the top and bottom of the antenna as 19° and 15° respectively. Determine the height of the aerial.

Answer 1

The height of the aerial is approximately 24.3 meters.

To determine the height of the aerial, we can use the trigonometric relationship between the angles of elevation and the height of the tower. Given that the surveyor stands 90m from the base of the tower and measures the angles of elevation to the top and bottom of the antenna as 19° and 15° respectively, we can set up the following equation:

`\tan(19^\circ) = \frac{\text{height of tower}}{90\text{m} + \text{height of antenna}}`

`\tan(15^\circ) = \frac{\text{height of tower}}{90\text{m}}`

By subtracting the second equation from the first, we can eliminate the height of the tower and solve for the height of the antenna:

`\tan(19^\circ) - \tan(15^\circ) = \frac{\text{height of tower}}{90\text{m} + \text{height of antenna}} - \frac{\text{height of tower}}{90\text{m}}`

Solving for the height of the antenna, we get:

`\text{height of antenna} = \frac{90\text{m} * (\tan(19^\circ) - \tan(15^\circ))}{\tan(19^\circ) - \tan(15^\circ)}`

Plugging in the values, we find:

`\text{height of antenna} \approx \frac{90\text{m} * (\tan(19^\circ) - \tan(15^\circ))}{\tan(19^\circ) - \tan(15^\circ)} \approx 24.3\text{m}`

Therefore, the height of the aerial is approximately 24.3 meters.