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Interactive Audio Lesson
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Introduction to the Problem
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Today, we're going to explore a practical example involving an electrician who needs to repair a fault on a pole. The pole stands at 5 meters tall. Can anyone tell me how we can find out the height the electrician needs to reach?
Isn't it 5 meters?
Not quite! She actually needs to reach 1.3 meters below the top. So, how will we find that height?
We can subtract 1.3 from 5!
Exactly! This gives us the height BD. So what is BD?
3.7 meters!
Right! BD equals 3.7 meters. Let's move onto finding the length of the ladder needed.
Using Trigonometric Ratios
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To solve for the ladder's length, we have to think about the triangle formed by the pole and the ladder. Which trigonometric ratio should we use, considering we know the angle is 60Β°?
We should use the sine function since we have the opposite side!
Exactly! Sine helps us relate the angle to the lengths. For sin(60Β°), we set up the equation as $\frac{BD}{BC} = \sin(60Β°)$. What does this imply for BC?
BC is the length of the ladder, right?
Correct! Using $\sin(60Β°)$, we can calculate BC. If BD is 3.7 m, what comes next?
We rearrange the equation to find BC!
Great job! Can you continue with the calculations?
Calculating the Ladder Length
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So we need to calculate BC. Recall that $\sin(60Β°) = \frac{\sqrt{3}}{2}$. Let's plug in the values. Do you remember how to rearrange?
Yeah! We can multiply both sides by BC to get $BC * \sin(60Β°) = BD$. Then we solve for BC.
Fantastic! Let's now apply it. What do we get if we calculate with $BD = 3.7$?
We have: $BC = \frac{3.7}{\sin(60Β°)}$ which leads to $BC = 4.28$ meters.
Perfect up to the approximated length of the ladder! Now, let's find out where the ladder's base should be placed.
Finding the Distance from the Pole
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To find the vertical distance from the pole, we should use cotangent. What is cot(60Β°) in terms of the triangle's sides?
It's $\frac{BD}{DC}$!
Good memory! Thus, we can express this as $DC = \frac{BD}{\cot(60Β°)}$. What would your calculation yield?
It gives us $DC = \frac{3.7}{0.577}$, which is approximately 2.14 meters.
Excellent! The foot of the ladder should be placed about 2.14 meters from the pole. Can someone summarize everything we've learned today?
We found the height to reach, calculated the ladder length using sine, and found out where to place the ladder using cotangent!
Introduction & Overview
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Quick Overview
Standard
This section explains a problem involving an electrician who needs to reach a point below the top of a pole. It demonstrates how to determine the length of a ladder needed when placed at an angle and the distance at which it should be positioned from the pole using trigonometric ratios.
Detailed
Detailed Summary
In this example, an electrician must repair an electrical fault on a pole height of 5 m. The task requires her to reach a point 1.3 m below the pole's top, making the required vertical reach 3.7 m (5 m - 1.3 m). To successfully calculate the necessary length of the ladder, the problem uses right-angled triangle properties and trigonometric ratios, specifically the sine function. The length of the ladder corresponds to the hypotenuse of the triangle formed.
The sine of the angle (60Β°) allows us to set up the equation using the formula:
$$\frac{BD}{BC} = \sin(60Β°)$$
Where BD (the vertical reach) equals 3.7 m, leading to the length of the ladder, BC, being approximately 4.28 m. Additionally, to find how far from the pole the ladder's base should be placed, the cotangent function provides the necessary calculation, yielding a distance of approximately 2.14 m. This example is significant for understanding practical applications of trigonometry in real-world scenarios.
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Audio Book
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Problem Statement
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An electrician has to repair an electric fault on a pole of height 5 m. She needs to reach a point 1.3m below the top of the pole to undertake the repair work. What should be the length of the ladder that she should use which, when inclined at an angle of 60Β° to the horizontal, would enable her to reach the required position? Also, how far from the foot of the pole should she place the foot of the ladder?
Detailed Explanation
The problem describes a situation where an electrician must reach a certain height on a pole to perform repairs. The pole is 5 meters tall, but she only needs to reach 3.7 meters high, which is calculated by subtracting the 1.3 meters from the total height. This leads us to the need to use a ladder inclined at a specific angle. The goal is to determine the length of the ladder as well as the distance from the base of the pole to where the ladder should be placed.
Examples & Analogies
Imagine a firefighter trying to reach a window on a tall building. Just like the firefighter needs to position their ladder at the right angle to reach the window safely, our electrician must also ensure her ladder is perfectly placed to safely access the area she needs to repair.
Calculating the Required Height
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BD = AD β AB = (5 β 1.3)m = 3.7 m.
Detailed Explanation
Here we use basic subtraction to find the height the electrician needs to reach (BD). The top of the pole (AD) is at 5 meters, and we subtract the 1.3 meters that she does not need to reach. This tells us that she needs to reach a height of 3.7 meters to complete her work.
Examples & Analogies
Think of it like climbing a staircase. If the staircase is 5 steps tall, and you're only required to climb to the 3rd step, then you aren't using the last 2 steps. This is exactly what we do with the pole β we only focus on the part we need.
Using Trigonometry to Find Ladder Length
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We need to find its length, i.e., the hypotenuse of the right triangle BDC. Here, BC represents the ladder. Now, can you think which trigonometric ratio should we consider? It should be sin 60Β°.
Detailed Explanation
Since we are dealing with a right triangle formed by the wall, the ground, and the ladder, we can apply trigonometric ratios. Specifically, we use the sine function because we know the angle (60Β°) and the opposite side (the height BD = 3.7 m). The function relates the opposite side to the hypotenuse (the length of the ladder).
Examples & Analogies
Imagine you are trying to reach something high on a shelf. If you know the angle you have to lean your ladder and how high the shelf is, trigonometry helps you find out how long the ladder needs to be to safely reach that height.
Calculating Ladder Length
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BD / BC = sin 60Β° or 3.7 / BC = 3 / 2. Therefore, BC = (3.7 * 2) / β3 = 4.28 m (approx.).
Detailed Explanation
Using the sine ratio, we set up the equation. By rearranging it, we can solve for the length of the ladder, BC. Multiplying gives us the length of the ladder required to reach the necessary height at the angle specified. This leads us to calculate approximately 4.28 meters as the length of the ladder.
Examples & Analogies
Just like you would measure a ladder to ensure it reaches your target height while being secure, here we calculate the exact ladder length she needs to safely perform her job.
Finding the Distance from the Pole
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Now, DC / BD = cot 60Β° = 1 / β3, i.e., DC = 3.7 / β3 = 2.14 m (approx.).
Detailed Explanation
Next, we need to determine where to place the base of the ladder, which involves another trigonometric relationship. Using cotangent, we once again relate the heights and distances involved. From our calculations, we find that the base of the ladder should be approximately 2.14 meters away from the pole.
Examples & Analogies
If you push your ladder too close to the wall, it might slip or become unstable. This calculation ensures that the ladder is placed at a safe distance so it remains stable when it's leaned against the pole, just like ensuring your ladder at home has firm footing.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Vertical Reach: The effective height that needs to be reached by the ladder.
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Trigonometric Ratios: Ratios that relate the angles and sides of a triangle, essential for calculations.
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Angle of Incline: The angle at which the ladder meets the ground, affecting the length needed.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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A ladder placed at a 60Β° angle can reach a height of 4.28 m when the vertical distance is 3.7 m.
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For a 5 m pole, the effective distance from the base of the ladder, given certain angles, can be calculated using cotangent.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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To climb up high and reach the sky, sin of sixty helps you fly!
π Fascinating Stories
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Imagine an electrician standing before a tall pole, needing her ladder to reach just the right spot. With a quick calculation, she finds the perfect angle, making her job safe and efficient.
π§ Other Memory Gems
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Remember: Sine = Opposite over Hypotenuse can help us find the height when the angle is clear!
π― Super Acronyms
LADDER
- Length Always Depends on Distance
- Elevation
- and Reach!
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Trigonometry
Definition:
A branch of mathematics dealing with the relationships between the angles and sides of triangles.
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Term: Sine Function
Definition:
A trigonometric function representing the ratio of the opposite side to the hypotenuse in a right triangle.
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Term: Cotangent Function
Definition:
A trigonometric function representing the ratio of the adjacent side to the opposite side in a right triangle.
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Term: Hypotenuse
Definition:
The longest side of a right triangle, opposite the right angle.