Finding the Height of a Tower Using Trigonometry

This problem involves using trigonometry, specifically the concept of the angle of elevation, to find the height of a tower given the height of an observer, the distance from the observer to the tower, and the angle of elevation from the observer's eye to the top of the tower.

Understanding the Geometry

We can visualize this scenario as a right-angled triangle. The base of the triangle is the distance from the observer to the tower. The perpendicular side is the height difference between the observer's eye level and the top of the tower. The angle of elevation is formed at the observer's eye, looking up towards the top of the tower.

• Observer's height = 2 m
• Distance from observer to tower = \(150\sqrt{3}\) m
• Angle of elevation from the observer's eye to the top of the tower = 60°

Let the total height of the tower be \(H\) meters. Since the angle of elevation is taken from the observer's eye level, we first need to find the height from the observer's eye level to the top of the tower. Let this height be \(h\) meters.

Applying Trigonometry to find the height difference

In the right-angled triangle formed:

• The opposite side to the angle of elevation (60°) is the height \(h\).
• The adjacent side to the angle of elevation (60°) is the distance from the observer to the tower, which is \(150\sqrt{3}\) m.

The trigonometric ratio relating the opposite and adjacent sides is the tangent function.

$$ \tan(\text{angle of elevation}) = \frac{\text{Opposite side}}{\text{Adjacent side}} $$

Substituting the given values:

$$ \tan(60\degree) = \frac{h}{150\sqrt{3}} $$

We know that the value of \(\tan(60\degree)\) is \(\sqrt{3}\).

$$ \sqrt{3} = \frac{h}{150\sqrt{3}} $$

To find \(h\), we multiply both sides of the equation by \(150\sqrt{3}\):

$$ h = \sqrt{3} \times 150\sqrt{3} $$

$$ h = 150 \times (\sqrt{3})^2 $$

$$ h = 150 \times 3 $$

$$ h = 450 \text{ m} $$

So, the height from the observer's eye level to the top of the tower is 450 m.

Calculating the Total Height of the Tower

The total height of the tower is the sum of the height from the ground to the observer's eye level (which is the observer's height) and the height from the observer's eye level to the top of the tower (\(h\)).

Total height of tower \(H\) = Observer's height + \(h\)

$$ H = 2 \text{ m} + 450 \text{ m} $$

$$ H = 452 \text{ m} $$

Therefore, the height of the tower is 452 meters.

Revision Table: Key Values and Calculation Steps

Additional Information on Angles of Elevation and Depression

The angle of elevation is the angle measured upwards from the horizontal line of sight to an object above the observer. Conversely, the angle of depression is the angle measured downwards from the horizontal line of sight to an object below the observer.

• Both angles are measured relative to a horizontal line.
• In problems involving a tall object and an observer, these angles are often used with trigonometric ratios (sine, cosine, tangent) in right-angled triangles.
• It is important to consider the height of the observer if the angle is measured from their eye level, as this height must be added to the calculated vertical distance to find the total height of the object.