Problem 55 - Algebra and Trigonometry

Question

Height of a Tower \(A\) water tower is located 325 \(\mathrm{ft}\) from a building (see the figure). From a window in the building, an observer notes that the angle of elevation to the top of the tower is \(39^{\circ}\) and that the angle of depression to the bottom of the tower is \(25^{\circ} .\) How tall is the tower? How high is the window?

Step-by-Step Solution

Answer

The tower is approximately 414.7 ft tall, and the window is 151.4 ft above the ground.

Step 1: Understanding the Problem

We have a right triangle scenario with the base distance of 325 ft. We are given two angles: one from the window to the top of the tower (angle of elevation = 39°) and one from the window to the bottom of the tower (angle of depression = 25°). We need to calculate the height above and below the window to find the height of the tower and the height of the window.

Step 2: Calculate Height Above the Window

Using the angle of elevation of 39° and the base of 325 ft, we apply the tangent function: \[ \tan(39°) = \frac{\text{Height above window}}{325} \]Solving for the height above the window, we get:\[ \text{Height above window} = 325 \times \tan(39°) \]

Step 3: Calculate Height Below the Window

For the angle of depression of 25°, we again use the tangent function by considering the same base of 325 ft:\[ \tan(25°) = \frac{\text{Height below window}}{325} \]Solving for the height below the window, we get:\[ \text{Height below window} = 325 \times \tan(25°) \]

Step 4: Determine Total Height of the Tower

Now, we add the height above the window to the height below the window to find the total height of the tower:\[ \text{Height of tower} = \text{Height above window} + \text{Height below window} \]

Step 5: Calculating Values and Conclusion

Calculating the values:\[ \text{Height above window} = 325 \times \tan(39°) \approx 263.3 \text{ ft} \]\[ \text{Height below window} = 325 \times \tan(25°) \approx 151.4 \text{ ft} \]Thus, the height of the tower is:\[ 263.3 + 151.4 = 414.7 \text{ ft} \]And, the height of the window is:\[ 151.4 \text{ ft} \]

Key Concepts

Angle of Elevation

Imagine you are standing straight and looking at the top of a tall object. Your line of sight forms an angle with the horizontal ground; this is known as the **angle of elevation**. It is always measured upwards, above the horizontal. In the context of trigonometry and right triangles, understanding this angle is crucial for calculating heights and distances.
In our problem, the observer in the building sees the top of the water tower from the window. The angle of elevation from the window to the tower’s top is 39°. This means there is a 39° angle formed by the line of sight upwards from the horizontal plane.

• Angle of elevation helps in determining the height of an object when its distance from the observer is known.
• It is a key concept when using trigonometric functions such as the tangent function.

By applying these principles, we are able to calculate how high above the window the top of the tower is.

Angle of Depression

The **angle of depression** is slightly different from the angle of elevation. It is the angle formed between the line of sight, when looking downward from a higher point to a lower point, and the horizontal line from the observer’s eye. It’s as if someone is looking down from a window to the ground.
In our exercise, the observer uses the window to look at the bottom of the water tower. This forms an angle of depression of 25°. This angle is crucial because it assists us in finding how far below the window the bottom of the tower is located.

• Angles of depression are essential for problems involving height differences when viewing from above.
• They also rely on trigonometric functions to calculate exact distances and heights.

Using the angle of depression correctly will help us accurately add to our total tower height calculation, by calculating the height below the window.

Tangent Function

The **tangent function** is one of the primary trigonometric functions, alongside sine and cosine. In right angle triangles, the tangent of an angle \( \theta \) is the ratio of the opposite side to the adjacent side: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
For our problem, since both angles of elevation and depression involve triangles with bases of 325 ft, the tangent function is particularly useful in calculating the unknown heights. We used it to find

• \(\tan(39°)\) gives the height above the window
• \(\tan(25°)\) gives the height below the window

Remember, the tangent function is perfect for such scenarios because it links an angle with linear dimensions (sides of the triangle), enabling distance and height calculations which are otherwise unseen.

Right Triangle

A **right triangle** is a triangle in which one of the angles is a right angle, which is exactly 90 degrees. This type of triangle is highly significant in trigonometry because it allows the use of specific functions like sine, cosine, and tangent to find unknown angles or sides.
In the context of this exercise, we are dealing with right triangles for both the top and bottom views from the window towards the water tower. Each forms a right triangle where:

• One side is the known distance from the building to the tower (325 ft).
• The angles (39° and 25°) help in forming right triangles, allowing us to use trigonometric formulas.

These triangles make it easier to employ the tangent function to determine the heights above and below the window, eventually leading to the total height of the water tower. Understanding the properties of right triangles is crucial for solving these types of problems efficiently.