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The length of the shadow of a vertical tower on level ground increases by 10 metres when the altitude of the sun changes from 45° to 30°. Then the height of the tower is (SSC CHSL 2014)
- a)5(√3 + 1) metres
- b)5(√3 - 1) metres
- c)5√3 metres
- d)5/√3 metres
Correct answer is option 'A'. Can you explain this answer?
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The length of the shadow of a vertical tower on level ground increases...
The length of the shadow of a vertical tower on level ground increases by 10 meters when the altitude of the sun changes from 45 degrees to 30 degrees.
Let's denote the original length of the shadow as x meters.
When the altitude of the sun is 45 degrees, we have a right triangle formed by the tower, the shadow, and the sun. The angle between the tower and the shadow is 90 degrees.
Using trigonometry, we can say that:
tan(45) = height of the tower / x
Since tan(45) = 1, we can simplify this to:
1 = height of the tower / x
Therefore, the height of the tower is equal to x meters.
When the altitude of the sun changes to 30 degrees, we have a similar right triangle formed by the tower, the longer shadow, and the sun. The angle between the tower and the longer shadow is 90 degrees.
Using trigonometry again, we can say that:
tan(30) = height of the tower / (x + 10)
Since tan(30) = 1/√3, we can simplify this to:
1/√3 = height of the tower / (x + 10)
Multiplying both sides by (x + 10), we get:
(x + 10)/√3 = height of the tower
Since we know that the height of the tower is equal to x, we can set up an equation:
(x + 10)/√3 = x
Multiplying both sides by √3, we get:
(x + 10)√3 = x√3
Expanding both sides, we get:
√3x + 10√3 = x√3
Subtracting √3x from both sides, we get:
10√3 = x(√3 - 1)
Dividing both sides by (√3 - 1), we get:
x = 10√3 / (√3 - 1)
Simplifying this expression, we get:
x ≈ 19.88
Therefore, the original length of the shadow is approximately 19.88 meters.
Let's denote the original length of the shadow as x meters.
When the altitude of the sun is 45 degrees, we have a right triangle formed by the tower, the shadow, and the sun. The angle between the tower and the shadow is 90 degrees.
Using trigonometry, we can say that:
tan(45) = height of the tower / x
Since tan(45) = 1, we can simplify this to:
1 = height of the tower / x
Therefore, the height of the tower is equal to x meters.
When the altitude of the sun changes to 30 degrees, we have a similar right triangle formed by the tower, the longer shadow, and the sun. The angle between the tower and the longer shadow is 90 degrees.
Using trigonometry again, we can say that:
tan(30) = height of the tower / (x + 10)
Since tan(30) = 1/√3, we can simplify this to:
1/√3 = height of the tower / (x + 10)
Multiplying both sides by (x + 10), we get:
(x + 10)/√3 = height of the tower
Since we know that the height of the tower is equal to x, we can set up an equation:
(x + 10)/√3 = x
Multiplying both sides by √3, we get:
(x + 10)√3 = x√3
Expanding both sides, we get:
√3x + 10√3 = x√3
Subtracting √3x from both sides, we get:
10√3 = x(√3 - 1)
Dividing both sides by (√3 - 1), we get:
x = 10√3 / (√3 - 1)
Simplifying this expression, we get:
x ≈ 19.88
Therefore, the original length of the shadow is approximately 19.88 meters.
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The length of the shadow of a vertical tower on level ground increases by 10 metres when the altitude of the sun changes from 45° to 30°. Then the height of the tower is (SSC CHSL 2014)a)5(√3 + 1) metresb)5(√3 - 1) metresc)5√3 metresd)5/√3metresCorrect answer is option 'A'. Can you explain this answer?
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