Understanding the Angle of Elevation Problem

This problem involves finding the required increase in the height of a tower based on changes in the angle of elevation from a fixed point on the ground. The angle of elevation is the angle formed by the line of sight from the observer to the top of an object and the horizontal line.

Given Information:

• Distance from the base of the tower to the observation point: 78 m
• Angle of elevation to the top of the unfinished tower: 30°
• Angle of elevation to the top of the finished tower: 60°

Using Trigonometry to Find Tower Heights

We can model this situation using right-angled triangles. The distance from the base is the adjacent side, and the height of the tower is the opposite side relative to the angle of elevation. The trigonometric ratio that relates the opposite and adjacent sides is the tangent function.

The formula is:

\(\tan(\text{angle of elevation}) = \frac{\text{Opposite side (Height)}}{\text{Adjacent side (Distance from base)}}\)

Step 1: Calculate the Height of the Unfinished Tower

Let \(h_1\) be the height of the unfinished tower.

\(\tan(30^\circ) = \frac{h_1}{78}\)

We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\).

So, \(\frac{1}{\sqrt{3}} = \frac{h_1}{78}\)

Solving for \(h_1\):

\(h_1 = 78 \times \frac{1}{\sqrt{3}}\)

\(h_1 = \frac{78}{\sqrt{3}}\)

To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{3}\):

\(h_1 = \frac{78}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{78\sqrt{3}}{3}\)

\(h_1 = 26\sqrt{3}\) m

Step 2: Calculate the Height of the Finished Tower

Let \(h_2\) be the height of the finished tower.

\(\tan(60^\circ) = \frac{h_2}{78}\)

We know that \(\tan(60^\circ) = \sqrt{3}\).

So, \(\sqrt{3} = \frac{h_2}{78}\)

Solving for \(h_2\):

\(h_2 = 78 \times \sqrt{3}\)

\(h_2 = 78\sqrt{3}\) m

Step 3: Calculate How Much Higher the Tower Must Be Raised

The height the tower must be raised is the difference between the height of the finished tower (\(h_2\)) and the height of the unfinished tower (\(h_1\)).

Height to be raised \( = h_2 - h_1\)

Height to be raised \( = 78\sqrt{3} - 26\sqrt{3}\)

Height to be raised \( = (78 - 26)\sqrt{3}\)

Height to be raised \( = 52\sqrt{3}\) m

Conclusion

The tower must be raised by \(52\sqrt{3}\) meters so that the angle of elevation of its top at the same point is 60°.

Revision Table: Key Trigonometric Values

Additional Information: Angles of Elevation and Depression

The angle of elevation is used when looking upwards at an object. The angle of depression is the angle formed by the horizontal line and the line of sight when looking downwards at an object. Both angles are measured from the horizontal line.

These concepts are fundamental in solving problems involving heights and distances using trigonometry. They often involve solving right-angled triangles using the sine, cosine, or tangent ratios, depending on the information given and what needs to be found.

Remembering the standard trigonometric values for common angles like 0°, 30°, 45°, 60°, and 90° is very helpful for solving these types of problems quickly.