A flagstaff is fixed at one corner of a squarefield of side 20 m.the angle of elevation to the top of flagstaff from the opposite corner of the field is found to be 30°.find the height of flagstaff
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A flagstaff is fixed at corner of square room of the side 20 m. The angle of elevation of the top of the flagstaff form opposite corner of the room is found to be 30 degree . Find the hight of the flagstaff with figuer.
Response
Tan30°=p/b
1/root3=p/30
p=17.32m
Well, let's see. If you imagine the square field as a giant game of hide and seek, and the flagstaff as the seeker standing tall in the corner, we can use a bit of trigonometry to solve this riddle.
Since the angle of elevation is 30°, we know that it forms a right triangle with the height of the flagstaff as the vertical leg, the distance from the opposite corner to the flagstaff as the hypotenuse, and the base as the side of the square field.
Now, if I remember correctly, the tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the flagstaff (let's call it h) and the adjacent side is half the length of the side of the square field, which is 20/2 = 10 m.
So, using the tangent of 30°, we have:
tan(30°) = h / 10
Now, let's solve for h:
h = 10 * tan(30°)
Using my high-tech calculations (aka Google), I found that tan(30°) is approximately 0.577.
So, plugging this in:
h = 10 * 0.577
h ≈ 5.77 m
So, according to my calculations, the height of the flagstaff is approximately 5.77 meters. That's one tall flagpole, isn't it?
To find the height of the flagstaff, we can use the concept of trigonometry.
Let's consider the square field. The distance from one corner to the opposite corner is equal to the diagonal of the square, which can be found using the Pythagorean theorem.
Given that the side of the square field is 20 m, the diagonal (d) can be found as:
d = √(20² + 20²)
d = √(400 + 400)
d = √800
d ≈ 28.28 m
Now, let's visualize the scenario. We have a right-angled triangle formed by the diagonal (d), the height of the flagstaff (h), and the distance between the flagstaff and the opposite corner of the square field.
In this triangle, the angle of elevation to the top of the flagstaff is 30°. The opposite side of this angle is equal to the height of the flagstaff (h), and the adjacent side is equal to half the length of the diagonal.
We can use the trigonometric ratio of tangent to find the height (h):
tan(angle) = opposite/adjacent
tan(30°) = h/(d/2)
Replacing the values:
tan(30°) = h/(28.28/2)
tan(30°) = h/14.14
√3 / 3 = h/14.14
Cross-multiplying:
h = (√3 / 3) * 14.14
h ≈ 8.17 m
Therefore, the height of the flagstaff is approximately 8.17 meters.
the field's diagonal has length 20√2
So, if the staff has height h, then
h/(20√2) = tan30°
Now just solve for h.