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The shadow of a tower standing on a level ground is found to be 40 metre longer when the suns altitude is 30° than when it is 60°. Find the length of the tower.
- a)20 √3 m
- b)10 m
- c)10 √3 m
- d)20 m
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
The shadow of a tower standing on a level ground is found to be 40 met...
We have CD = 40 m.
In ΔABC,
tan 60 = AB/BC
⇒ √3 = AB/BC
⇒ AB = √3 BC
Now, we have
In ΔABD,
⇒ tan 30 = AB/(BC + 40)
⇒ BC + 40 = 3 BC
⇒ BC = 20 m.
Hence, AB = √3 BC
⇒ AB = 20√3 m.
Most Upvoted Answer
The shadow of a tower standing on a level ground is found to be 40 met...
Understanding the Problem
We need to find the height of a tower given the lengths of its shadow at two different altitudes of the sun.
Given Information
- Shadow length difference: 40 meters
- Sun altitude (Angle of elevation):
- 30 degrees
- 60 degrees
Using Trigonometry
We can use the tangent function from trigonometry, where:
- tan(angle) = height of the tower / length of the shadow
Let the height of the tower be 'h' meters.
Calculating Shadow Lengths
1. At 60 degrees:
- tan(60) = h / L1
- L1 = h / √3
2. At 30 degrees:
- tan(30) = h / L2
- L2 = h
Setting Up the Equation
According to the problem:
L2 = L1 + 40
Substituting the expressions for L1 and L2:
h = (h / √3) + 40
Solving for Height
Now, multiply through by √3 to eliminate the fraction:
√3 * h = h + 40√3
Rearranging gives:
√3 * h - h = 40√3
(√3 - 1)h = 40√3
Now, we can solve for h:
h = 40√3 / (√3 - 1)
Calculating Height
To find a numerical value, we can rationalize the denominator:
h = 40√3 * (√3 + 1) / (3 - 1) = 20√3
Conclusion
Thus, the height of the tower is:
20√3 m
So, the correct answer is option A: 20√3 m.
We need to find the height of a tower given the lengths of its shadow at two different altitudes of the sun.
Given Information
- Shadow length difference: 40 meters
- Sun altitude (Angle of elevation):
- 30 degrees
- 60 degrees
Using Trigonometry
We can use the tangent function from trigonometry, where:
- tan(angle) = height of the tower / length of the shadow
Let the height of the tower be 'h' meters.
Calculating Shadow Lengths
1. At 60 degrees:
- tan(60) = h / L1
- L1 = h / √3
2. At 30 degrees:
- tan(30) = h / L2
- L2 = h
Setting Up the Equation
According to the problem:
L2 = L1 + 40
Substituting the expressions for L1 and L2:
h = (h / √3) + 40
Solving for Height
Now, multiply through by √3 to eliminate the fraction:
√3 * h = h + 40√3
Rearranging gives:
√3 * h - h = 40√3
(√3 - 1)h = 40√3
Now, we can solve for h:
h = 40√3 / (√3 - 1)
Calculating Height
To find a numerical value, we can rationalize the denominator:
h = 40√3 * (√3 + 1) / (3 - 1) = 20√3
Conclusion
Thus, the height of the tower is:
20√3 m
So, the correct answer is option A: 20√3 m.
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