Question

Two vertical towers have heights \(9 \mathrm{~m}\) and \(17.2 \mathrm{~m}\) and are \(42 \mathrm{~m}\) apart. (a) Calculate the angle of elevation from the base of the shorter tower to the top of the taller tower. (b) Calculate the angle of depression from the top of the taller tower to the top of the shorter tower.

Short answer

Question: Calculate the angle of elevation from the base of the shorter tower to the top of the taller tower and the angle of depression from the top of the taller tower to the top of the shorter tower, given that the height of the shorter tower is 9m, the height of the taller tower is 17.2m, and the distance between the towers is 42m. Answer: The angle of elevation from the base of the shorter tower to the top of the taller tower is approximately 11.04°, and the angle of depression from the top of the taller tower to the top of the shorter tower is approximately 12.16°.

Step-by-step solution

Draw the diagram

First, we will draw the diagram representing the given scenario. From the base of the shorter tower, draw a horizontal line segmentABCDE, making an angle with the vertical shorter tower. Draw the vertical taller tower parallel to the shorter tower, with a distance of 42m between them. Label the base of the shorter tower as A, top of the shorter tower as B, the base of the taller tower as C, the top of the taller tower as D, and the intersection of the horizontal line segment and the taller tower as E.

Identify the given information and the unknowns

The given information is as follows: - Height of the shorter tower, AB = 9m - Height of the taller tower, CD = 17.2m - Distance between the towers, AC = 42m The unknowns are: - Angle of Elevation from A to D, ∠BAD - Angle of Depression from D to B, ∠DBE

Calculate height difference and find triangles

In order to find the angle of elevation and depression, we first need to calculate the height difference between the two towers. Height difference, DE = CD - AB = 17.2 - 9 = 8.2m Now, we can identify two right-angled triangles: 1. Right Triangle ΔABE with AB as one leg, BE as the other leg, and AE as the hypotenuse. 2. Right Triangle ΔACD with AD as one leg, AC as the other leg, and AE as the hypotenuse.

Calculate the angle of elevation

In right triangle ΔACD, we will use the tangent trigonometric ratio to determine the angle of elevation ∠BAD. tan(∠BAD) = \frac{opposite \ side}{adjacent \ side} = \frac{AD}{AC} So, tan(∠BAD) = \frac{8.2}{42} Now, calculate the angle ∠BAD: ∠BAD = arctan(\frac{8.2}{42}) Therefore, the angle of elevation is ∠BAD ≈ 11.04°.

Calculate the angle of depression

In right triangle ΔABE, we will use the tangent trigonometric ratio to determine the angle of depression ∠DBE. tan(∠DBE) = \frac{opposite \ side}{adjacent \ side} = \frac{BE}{AC} But, BE = AB, So, tan(∠DBE) = \frac{9}{42} Now, calculate the angle ∠DBE: ∠DBE = arctan(\frac{9}{42}) Therefore, the angle of depression is ∠DBE ≈ 12.16°. The final answers are: (a) The angle of elevation from the base of the shorter tower to the top of the taller tower is approximately 11.04°. (b) The angle of depression from the top of the taller tower to the top of the shorter tower is approximately 12.16°.

Key concepts

Understanding the Angle of Elevation

The angle of elevation is an important concept in trigonometry, often used when analyzing the height of objects from a certain point on the ground level. The angle is measured from the horizontal. Imagine standing at the base of a short tower and looking up at a taller neighboring tower. The angle your line of sight forms with a flat horizontal line at your eye level is the angle of elevation.

Here's how you can think about the angle of elevation:

• It's always considered with respect to a horizontal line.
• A larger angle of elevation suggests that the object you are observing is either closer or much taller.
• Understanding angles of elevation can help in calculating distances and heights in various real-world problems.

To calculate the angle of elevation, you can use trigonometric ratios, specifically the tangent ratio. Suppose you observe an object vertically above a certain point, the angle of elevation can be calculated using the formula:\[ \tan(\text{Angle}) = \frac{\text{Opposite Side}}{\text{Adjacent Side}} \]In our original problem, it involves looking up from the top of a shorter tower to the top of a taller one.

Exploring the Angle of Depression

The angle of depression is a concept that is less about looking up and more about looking down. It represents the angle between your line of sight and a horizontal reference line when you're viewing an object situated below your level.

When you are atop the taller tower looking over to the shorter one, the angle formed by your sight line against the horizontal is the angle of depression.

• This angle is always outside of the triangle formed with the object being observed.
• Just like the angle of elevation, the angle of depression can be large or small, depending on the distance and height differences involved.
• In simpler terms, it helps determine how far beneath an object is compared to your own standing point.

To find the angle of depression, the same trigonometric principle applies: the tangent ratio. For example, if you're determining how steep a slope is from a tall building, it can effectively be represented as:\[ \tan(\text{Angle}) = \frac{\text{Opposite Side}}{\text{Adjacent Side}} \]In our given scenario, we are looking from the top of the taller tower down to the top of the shorter one.

Right-Angled Triangles in Trigonometry

Right-angled triangles are a central theme in trigonometry. They are special because one of their angles is exactly 90 degrees. This property makes calculations simpler by allowing the use of trigonometric ratios like sine, cosine, and tangent.

In our scenario with the two towers and their respective heights, the concept of right-angled triangles is crucial to solving for the angles of elevation and depression.

• These triangles help us understand the relationship between different sides and angles.
• Applications of right-angled triangles go beyond physics and engineering; they're also standard problem-solvers in everyday situations like construction and navigation.
• The sides of a right-angled triangle include: the hypotenuse (the longest side), and two legs (one adjacent and one opposite to the angle in question).

The economy and elegance of calculating with right-angled triangles stem from trigonometric functions—especially tangent in our context:\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]Thus, understanding right-angled triangles assists greatly in applying trigonometry to real-world problems.