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9.1.7 - Example 7

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Understanding the Problem

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Teacher
Teacher

Today, we're going to explore how to determine distances using angles of depression. Can anyone remind me what an angle of depression is?

Student 1
Student 1

Isn't it the angle below the horizontal line of sight?

Teacher
Teacher

Exactly! Now, we have a bridge that's 3 meters high. We'll be looking at two angles of depression from this height: 30° and 45°. How do you think we can find the width of the river using these angles?

Student 2
Student 2

We can use right triangles since we know the height and the angles!

Teacher
Teacher

Correct! We'll create two right triangles based on these angles. Let's delve into how to calculate the distances.

Calculating Distances

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Teacher
Teacher

Using triangle APD where ∠A is 30°, what can we say about the relationship between the sides?

Student 3
Student 3

We can use the tangent function for tan(30°) = PD/AD, right?

Teacher
Teacher

Exactly! Is anyone able to calculate AD from that formula?

Student 4
Student 4

Yes! It's AD = PD * √3. If PD is 3 m, then AD = 3√3 m.

Teacher
Teacher

Very well done! Now let’s move to triangle PBD. How can we use the angle of 45°?

Combining Calculations

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Teacher
Teacher

Now we have AD calculated. For triangle PBD with an angle of 45°, what can we conclude?

Student 1
Student 1

In that case, BD should equal PD because tan(45°) = 1!

Teacher
Teacher

Correct again! BD = 3 m. Now, who can tell me the total width of the river AB?

Student 3
Student 3

It would be AB = AD + BD which is 3 + 3√3 m!

Teacher
Teacher

Exactly! That gives us the final width. Great teamwork, everyone!

Revisiting Concepts

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Teacher
Teacher

Before we finish, let’s recap how we solved the problem. What steps did we take?

Student 4
Student 4

We started by identifying angles of depression!

Teacher
Teacher

Yes! And then we set up our triangles using tangent ratios based on those angles. But what’s special about the angle of 45°?

Student 2
Student 2

It makes the two sides equal in triangle PBD!

Teacher
Teacher

Correct! This is an important property of 45° angles. Let's remember these points for future geometry problems.

Introduction & Overview

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Quick Overview

This section involves calculating the width of a river using angles of depression from a bridge.

Standard

In Example 7, students learn how to determine the width of a river based on angles of depression observed from a bridge. The example illustrates the application of trigonometric ratios in right triangles, specifically using angles of depression and the height of the bridge.

Detailed

Example 7 Analysis

In this example, a person stands on a bridge that is 3 meters high and records the angles of depression to the banks of a river on both sides. The angles of depression are 30° and 45°, respectively. By applying the principles of right triangle trigonometry, we can determine the width of the river by calculating the horizontal distances to each bank from the foot of the bridge.

Definitions and Trigonometric Relationships Used:
- Angle of Depression: The angle formed by a horizontal line and a line of sight down to an object below the line.
- Tan Function: Utilized in the right triangles formed by the height of the bridge and the distances to the banks.

In the analysis:
1. Triangle APD uses angle 30° to calculate distance AD.
2. Triangle PBD uses angle 45° to find distance BD.

Finally, the total width of the river AB is the sum of AD and BD, yielding the final result in the form of 3(1 + √3)m.

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Audio Book

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Understanding the Problem

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From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30° and 45°, respectively. If the bridge is at a height of 3 m from the banks, find the width of the river.

Detailed Explanation

In this problem, we need to determine the width of a river by using the angles of depression from a bridge. The angles given are 30° and 45°, which means that from the point on the bridge, we can visualize two triangles formed between the bridge and the banks of the river. The height from the bridge to the banks is 3 meters. This information helps us set up our calculations.

Examples & Analogies

Imagine you are standing on a bridge looking down at two banks of a river. One bank is 30° to the left, and the other is 45° to the right. Just like using a ruler to measure, we will use our angles and height to find out how wide the river is, just like measuring the distance across a playground.

Setting Up the Triangles

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In Fig 9.10, A and B represent points on the bank on opposite sides of the river, so that AB is the width of the river. P is a point on the bridge at a height of 3 m, i.e., DP = 3 m. We are interested to determine the width of the river, which is the length of the side AB of the triangle APB.

Detailed Explanation

In this chunk, we define key points in the problem: points A and B represent the banks of the river, while point P is where we stand on the bridge. We need to find the length AB, which is the width of the river. To do this, we'll analyze two right triangles—one with angle 30° (triangle APD) and the other with angle 45° (triangle PBD). These triangles give us a way to use the heights and angles to find the distances we need.

Examples & Analogies

Think of this setup like a seesaw on a playground. You stand on one side at a height, and underneath you, two friends are standing on opposite sides lower down. As you look down at them, the angles created by your gaze give you clues about how far apart they are, similar to how we are looking from the bridge at the riverbanks.

Calculating Distances

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Now, AB = AD + DB. In right Δ APD, ∠ A = 30°. So, tan 30° = PD / AD, i.e., 1/√3 = 3/AD or AD = 3√3 m. Also, in right Δ PBD, ∠ B = 45°. So, BD = PD = 3 m.

Detailed Explanation

Here, we break down our triangle calculations. We first find AD, where we use the tangent function. The angle 30° leads us to use the tangent ratio which gives us AD = 3√3 m. For triangle PBD, since it's a 45° angle, we find BD equals PD, which is straightforward since both are 3 m. Now that we have both AD and BD, we can combine them to find the total width AB.

Examples & Analogies

Imagine you’re measuring distances with a measuring tape on those same friends from the seesaw. First, you measure one friend (AD), and it turns out to be a bit longer because you're looking from a greater angle, and then you measure the other friend (BD) straight down, and it’s the same as your height! Combining these will tell you how far they are apart, similar to adding our distances to find the river width.

Final Calculation of the Width

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Now, AB = BD + AD = 3 + 3√3 = 3(1 + √3) m. Therefore, the width of the river is 3(√3 + 1) m.

Detailed Explanation

Finally, we combine our found values for AD and BD to find the width of the river, represented as AB. The final expression simplifies the width calculation into a clear formula, indicating that the total width of the river is 3 times the sum of 1 and the square root of 3. This result is crucial for understanding the overall distance we need.

Examples & Analogies

It's like putting together parts of a jigsaw puzzle. Once you measured both distances of your friends, putting them together gives you the whole picture of how wide the playground is! Here, the final width helps us visualize how wide that river really is.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Angle of Depression: The angle formed with a horizontal line when looking down to an object below.

  • Height of the Bridge: The vertical distance from the bridge to the ground or water level.

  • Tan Function: A ratio used in trigonometry to relate angles and lengths of triangles.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Consider a building that is 10 meters tall. If the angle of depression to the ground from the top of the building is 60°, what is the distance from the base of the building to the point directly below the top of the building?

  • From a hill 20 m high, if the angle of depression to a car is 30°, calculate the distance from the base of the hill to the car.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • If you see the bank, look down the plank; at thirty degrees you’ll see the trees!

📖 Fascinating Stories

  • Imagine a bridge where a person looks down, seeing a beautiful river flowing below. The angles guide them to calculate the distance with ease, like a treasure hunt under the trees.

🧠 Other Memory Gems

  • When calculating lengths, remember: TAD (Tangent for Angle and Distance).

🎯 Super Acronyms

Use `TAD` to remember Tangent, Angle, Distance – the key to solving!

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Angle of Depression

    Definition:

    The angle formed by a horizontal line and a line of sight from above to a point below the horizontal.

  • Term: Height

    Definition:

    The measure of how tall something is from its base to its top, here referring to the height of the bridge.

  • Term: Tan Function

    Definition:

    A trigonometric function that relates the angle of a right triangle to the ratio of the opposite side to the adjacent side.