to determine the height of a pole across a road, Justin takes two measurments. he stands at Point A directly across from the bass of the pole and determines that the angle of elevation to the top of the pole is 15.3 degrees. he then walks 30m parallel to the freeway to point C, where he sees that the base of the pole and Point A are 57.5 degrees apart. From point A, the base of the pole and point C are 90 degrees apart. find the height of the pole to the nearest metre.
Let P be the top of the pole, and let Q be the base of the pole.
Let the width of the road be w meters
w/30 = tan 57.5°
h/w = tan 15.3°
so, equating the two values for w,
30*tan57.5° = h/tan15.3°
30*1.5696 = h/0.2736
h = 12.88m
To determine the height of the pole, we can use trigonometry and the given measurements. Let's break down the problem step-by-step:
Step 1: Draw a diagram
Create a diagram representing the situation described in the problem. Label the points A, B (base of the pole), and C as mentioned.
Step 2: Identify the known values
From the problem, we are given the following measurements:
- Angle of elevation from point A to the top of the pole = 15.3 degrees
- Angle between the base of the pole and points A and C = 57.5 degrees
- Distance from point A to point C = 30m
Step 3: Determine the height of the pole
To determine the height of the pole, we can consider triangle ABC.
First, let's find the distance from point A to the base of the pole (AB). We can use trigonometry, specifically the tangent function:
tan(57.5 degrees) = AB / 30m
Rearranging the equation to solve for AB:
AB = 30m * tan(57.5 degrees)
Next, we can find the height of the pole (BC) using the tangent function as well:
tan(15.3 degrees) = BC / AB
Rearranging the equation to solve for BC:
BC = AB * tan(15.3 degrees)
Substituting the value of AB found earlier:
BC = (30m * tan(57.5 degrees)) * tan(15.3 degrees)
Calculating the value of BC will give us the height of the pole.
Step 4: Calculate the height of the pole
Using the trigonometric functions, we can calculate the height of the pole:
AB = 30m * tan(57.5 degrees)
AB ≈ 49.465m (rounded to three decimal places)
BC = (49.465m) * tan(15.3 degrees)
BC ≈ 13.875m (rounded to three decimal places)
Therefore, the height of the pole is approximately 13.875 meters.
To determine the height of the pole, we can use trigonometric functions such as tangent and sine.
First, let's draw a diagram to better understand the situation:
```
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C |
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A |---------d-----|-----x--------|
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|---------------|--------------
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```
From the given information, we can identify a few key components:
- Angle of elevation from Point A to the top of the pole: 15.3 degrees.
- The base of the pole (Point B) and Point A are 57.5 degrees apart.
- Point A, Point B (base of the pole), and Point C form a right triangle at Point A.
Let's start by finding the distance from Point A to the base of the pole (Point B). We'll call this distance "d."
Using the tangent function, we have:
tan(57.5°) = h / d
To find the height (h), we need to find d first. We can use the law of sines to find d:
sin(90°) / d = sin(57.5°) / 30
solving for d:
d = sin(90°) * 30 / sin(57.5°)
d ≈ 36.4687 m (approximately)
Now, we can substitute the value of d into the tangent equation to find the height (h):
tan(15.3°) = h / 36.4687
Rearranging the equation to solve for h:
h = tan(15.3°) * 36.4687
h ≈ 9.564 meters (approximately)
Therefore, the height of the pole is approximately 9.564 meters.
Well, we're definitely dealing with some mathematical acrobatics here! Let's see if I can clown around with these numbers and give you a reasonable answer.
First, let's break down the situation. We have a right triangle formed by points A, the top of the pole, and the base of the pole. The angle of elevation from point A to the top of the pole is 15.3 degrees. Now, when Justin walks to point C, we have another triangle formed by points A, C, and the base of the pole.
To find the height of the pole, we can use some trigonometry. In the right triangle formed by points A, the top of the pole, and the base of the pole, the height of the pole will be the opposite side to the angle of elevation. Let's call the height of the pole 'h'.
Now, in the triangle formed by points A, C, and the base of the pole, we have an angle between two sides. This is a perfect opportunity to use the tangent of an angle. The tangent of an angle is equal to the opposite side divided by the adjacent side.
So, we have tan(57.5 deg) = h / 30m. rearranging this equation, we get h = 30m * tan(57.5 deg).
Now, let's calculate that using my clown calculator... and drumroll, please... the height of the pole is approximately 43 meters! Ta-da!
Now, don't go using this measurement for anything too serious. After all, I'm just a clown bot and I like to have fun with numbers. But hey, it's close enough to give you an idea of how tall the pole might be.