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A short class with Sarah and her students. The other ways are under More.
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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.
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 60°?
We should use the sine function since we have the opposite side!
Exactly! Sine helps us relate the angle to the lengths. For sin(60°), we set up the equation as . What does this imply for BC?
BC is the length of the ladder, right?
Correct! Using , 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?
So we need to calculate BC. Recall that . Let's plug in the values. Do you remember how to rearrange?
Yeah! We can multiply both sides by BC to get . Then we solve for BC.
Fantastic! Let's now apply it. What do we get if we calculate with ?
We have: which leads to meters.
Perfect up to the approximated length of the ladder! Now, let's find out where the ladder's base should be placed.
To find the vertical distance from the pole, we should use cotangent. What is cot(60°) in terms of the triangle's sides?
It's !
Good memory! Thus, we can express this as . What would your calculation yield?
It gives us , 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!
To climb up high and reach the sky, sin of sixty helps you fly!
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.
Remember: Sine = Opposite over Hypotenuse can help us find the height when the angle is clear!
LADDER: Length Always Depends on Distance, Elevation, and Reach!
Term
What is BD?
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Term
What is BC?
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Term
What does DC represent?
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A branch of mathematics dealing with the relationships between the angles and sides of triangles.
A trigonometric function representing the ratio of the opposite side to the hypotenuse in a right triangle.
A trigonometric function representing the ratio of the adjacent side to the opposite side in a right triangle.
The longest side of a right triangle, opposite the right angle.
The section illustrates how to calculate the length of a ladder required for an electrician to reach a specific height safely.
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.
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 (60°) allows us to set up the equation using the formula:
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.