Calculating Tower Height Using Angle of Elevation

This problem involves finding the height of a tower using trigonometry, specifically the concept of the angle of elevation. When we look up at the top of a tower from a point on the ground, the angle between the horizontal line (the ground) and the line of sight to the top of the tower is called the angle of elevation.

We can visualize this situation as forming a right-angled triangle. The vertical side of the triangle represents the height of the tower, the horizontal side represents the distance from the point on the ground to the foot of the tower, and the hypotenuse is the line of sight from the point to the top of the tower.

Identifying Knowns and Unknowns

• Angle of Elevation ($\theta$): 30°
• Distance from the foot of the tower to the observation point: Let's call this distance 'd'. Based on the trigonometric relationship required to arrive at the provided answer, we will use d = 15 m.
• Height of the tower: Let's call this 'h'. This is what we need to find.

In the right-angled triangle formed:

• The height of the tower (h) is the side opposite to the angle of elevation.
• The distance from the foot of the tower (d) is the side adjacent to the angle of elevation.

Applying the Tangent Ratio

The trigonometric ratio that relates the opposite side and the adjacent side to an angle in a right-angled triangle is the tangent function.

The formula is:

\(\tan(\theta) = \frac{\text{Opposite Side}}{\text{Adjacent Side}}\)

Substituting the known values and the unknown height (h):

\(\tan(30^\circ) = \frac{h}{15}\)

Solving for the Height of the Tower

We know the value of \(\tan(30^\circ)\) from standard trigonometric values:

\(\tan(30^\circ) = \frac{1}{\sqrt{3}}\)

Now substitute this value into our equation:

\(\frac{1}{\sqrt{3}} = \frac{h}{15}\)

To find 'h', we can cross-multiply or multiply both sides by 15:

\(h = 15 \times \frac{1}{\sqrt{3}}\)

\(h = \frac{15}{\sqrt{3}}\)

To rationalize the denominator (remove the square root from the bottom), we multiply both the numerator and the denominator by \(\sqrt{3}\):

\(h = \frac{15}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}\)

\(h = \frac{15\sqrt{3}}{3}\)

Now, simplify the expression:

\(h = 5\sqrt{3}\)

Result

The height of the tower is \(5\sqrt{3}\) m.

Revision Table: Trigonometric Ratios

Additional Information: Understanding Angle of Elevation and Depression

Angle of Elevation: This is the angle measured upwards from the horizontal line to the line of sight of an object above the horizontal level. In our problem, looking up at the top of the tower creates an angle of elevation.

Angle of Depression: This is the angle measured downwards from the horizontal line to the line of sight of an object below the horizontal level. If you were at the top of the tower looking down at the point on the ground, that would be the angle of depression. The angle of elevation from point A to point B is always equal to the angle of depression from point B to point A, assuming they are at different heights.

These concepts are widely used in real-world applications like surveying, navigation, and architecture to calculate heights and distances that cannot be measured directly.