Solving Tower Height using Angle of Elevation

This problem involves using trigonometry to find the height of a tower, given the height of a pole, the distance between the pole and the tower, and the angle of elevation from the top of the pole to the top of the tower. We can visualize this scenario as forming a right-angled triangle.

Understanding the Setup

• We have an upright pole with a height of \(24\sqrt{3}\) feet.
• We have an upright tower whose height we need to find.
• The horizontal distance between the foot of the pole and the foot of the tower is 60 feet.
• The angle of elevation from the top of the pole to the top of the tower is \(60^\circ\).

Let's denote the height of the tower as \(H\). The key to solving this problem is to consider the right triangle formed by:

1. The horizontal line segment from the top of the pole, parallel to the ground, extending towards the tower. The length of this segment is equal to the distance between the foot of the pole and the foot of the tower, which is 60 feet. This forms the adjacent side of our triangle.
2. The vertical line segment from the top of the tower down to the level of the top of the pole. This represents the difference in height between the tower and the pole, i.e., \(H - 24\sqrt{3}\). This forms the opposite side of our triangle.
3. The line of sight from the top of the pole to the top of the tower, which is the hypotenuse, and forms the \(60^\circ\) angle of elevation with the horizontal line (segment 1).

Applying Trigonometry (SOH CAH TOA)

In the right-angled triangle described above, we have the angle of elevation (\(60^\circ\)), the adjacent side (60 feet), and the opposite side (\(H - 24\sqrt{3}\)). The trigonometric ratio that relates the opposite side and the adjacent side is the tangent function.

The tangent of an angle is defined as:

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

In our case, \(\theta = 60^\circ\), the opposite side is \(H - 24\sqrt{3}\), and the adjacent side is 60 feet.

So, we can write the equation:

\(\tan(60^\circ) = \frac{H - 24\sqrt{3}}{60}\)

Solving for the Tower Height

We know the value of \(\tan(60^\circ)\) from standard trigonometric values. \(\tan(60^\circ) = \sqrt{3}\).

Substituting this value into the equation:

\(\sqrt{3} = \frac{H - 24\sqrt{3}}{60}\)

To find \(H\), we first multiply both sides of the equation by 60:

\(60 \times \sqrt{3} = H - 24\sqrt{3}\)

\(60\sqrt{3} = H - 24\sqrt{3}\)

Now, add \(24\sqrt{3}\) to both sides of the equation to isolate \(H\):

\(60\sqrt{3} + 24\sqrt{3} = H\)

Combine the terms on the left side:

\((60 + 24)\sqrt{3} = H\)

\(84\sqrt{3} = H\)

So, the height of the tower is \(84\sqrt{3}\) feet.

Comparing this result with the given options, we find that it matches option 1.

Revision Table: Key Concepts

Additional Information: Angle of Elevation and Depression

The angle of elevation is always measured upwards from a horizontal line to the object. Conversely, the angle of depression is measured downwards from a horizontal line to the object. In problems involving two objects at different heights, understanding which angle is given (elevation or depression) and from which point it is measured is crucial for setting up the correct right triangle and trigonometric equation.

In this problem, the horizontal distance of 60 feet connects the bases. However, the angle of elevation is measured from the top of the pole. Therefore, the 60 feet horizontal distance is applied to the level of the top of the pole, and the vertical side of the triangle is the difference in height between the tower and the pole.

Remembering the standard values for sine, cosine, and tangent for \(30^\circ\), \(45^\circ\), and \(60^\circ\) is very helpful for solving many trigonometry problems quickly.