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Two men are on the same side of a tower. They observe the angles of elevation of the top of the tower as 30° and 45° respectively. If the height of the tower is 100m, then the distance between them is
- a)100(1−√3)m
- b)100(√3+1)m
- c)100(√3−1)m
- d)none of these.
Correct answer is option 'C'. Can you explain this answer?
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Degrees and 45 degrees, respectively. If the distance between the two men is 100 meters, how tall is the tower?
Let's denote the height of the tower as "h".
From the first man's perspective, the angle of elevation of the top of the tower is 30 degrees. This means that the opposite side is h and the adjacent side is x (the distance between the first man and the tower).
Using the tangent function, we can write:
tan(30) = h / x
From the second man's perspective, the angle of elevation of the top of the tower is 45 degrees. This means that the opposite side is h and the adjacent side is (100 - x) (the remaining distance between the second man and the tower).
Using the tangent function, we can write:
tan(45) = h / (100 - x)
Now we have a system of equations:
1) tan(30) = h / x
2) tan(45) = h / (100 - x)
To solve this system, we can eliminate h by setting the two equations equal to each other:
h / x = h / (100 - x)
Cross-multiplying, we get:
h(100 - x) = hx
Expanding and simplifying:
100h - hx = hx
Bringing all the "hx" terms to one side:
100h = 2hx
Dividing both sides by 2h:
50 = x
Now we can substitute this value of x into one of the original equations (let's use the first one) to solve for h:
tan(30) = h / 50
Using the value of the tangent of 30 degrees (approximately 0.577):
0.577 = h / 50
Cross-multiplying:
h = 0.577 * 50
Simplifying:
h = 28.85
Therefore, the height of the tower is approximately 28.85 meters.
Let's denote the height of the tower as "h".
From the first man's perspective, the angle of elevation of the top of the tower is 30 degrees. This means that the opposite side is h and the adjacent side is x (the distance between the first man and the tower).
Using the tangent function, we can write:
tan(30) = h / x
From the second man's perspective, the angle of elevation of the top of the tower is 45 degrees. This means that the opposite side is h and the adjacent side is (100 - x) (the remaining distance between the second man and the tower).
Using the tangent function, we can write:
tan(45) = h / (100 - x)
Now we have a system of equations:
1) tan(30) = h / x
2) tan(45) = h / (100 - x)
To solve this system, we can eliminate h by setting the two equations equal to each other:
h / x = h / (100 - x)
Cross-multiplying, we get:
h(100 - x) = hx
Expanding and simplifying:
100h - hx = hx
Bringing all the "hx" terms to one side:
100h = 2hx
Dividing both sides by 2h:
50 = x
Now we can substitute this value of x into one of the original equations (let's use the first one) to solve for h:
tan(30) = h / 50
Using the value of the tangent of 30 degrees (approximately 0.577):
0.577 = h / 50
Cross-multiplying:
h = 0.577 * 50
Simplifying:
h = 28.85
Therefore, the height of the tower is approximately 28.85 meters.
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