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From the top and the foot of a house h metres high the angles of elevation of the top of a tower are β and α respectively, then the height of the tower is
  • a)
    [(h sinβ)/(cosβ-sinα)]
  • b)
    [(h cosβ)/(cosβ-cosα)]
  • c)
    [(h tanβ)/(tanβ-tanα)]
  • d)
    [(h cotβ)/(cotβ-cotα)]
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
From the top and the foot of a house h metres high the angles of eleva...
Explanation:

- Let the height of the tower be x meters.
- From the top of the house, the angle of elevation to the top of the tower is β.
- From the foot of the house, the angle of elevation to the top of the tower is α.

Using Trigonometry:

- From the top of the house, we have:
- tan β = x / h (1)

- From the foot of the house, we have:
- tan α = x + h / h (2)

- Solving equations (1) and (2) simultaneously, we get:
- x = h(tan β - tan α) / (1 - tan α tan β)
- x = h(tan β - tan α) / (tan β - tan α)
- x = h tan β / (tan β - tan α)

Therefore, the height of the tower is given by:
Height of the tower = h tan β / (tan β - tan α)

So, the correct answer is C: [(h tanβ)/(tanβ-tanα)].
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Most Upvoted Answer
From the top and the foot of a house h metres high the angles of eleva...
Given:
The height of the house = h meters
Angles of elevation to the top of the tower from the top and the foot of the house = β and α respectively

Let's solve step by step:

Step 1: Determine the height of the tower using trigonometry
Let x be the height of the tower
From the top of the house,
tan β = x/h ...(1)
From the foot of the house,
tan α = (x + h)/h = x/h + 1 ...(2)

Step 2: Solve for x
From (1), x = h tan β
Substitute x in (2),
tan α = h tan β/h + 1
tan α = tan β + 1
tan β = tan α - 1
tan β - tan α = -1
tan β/(1 - tan α) = -1
tan β = -1 + tan α
tan β - tan α = tan β tan α
tan β - tan α = (sin β cos α - cos β sin α)/(cos β cos α)
tan β - tan α = (sin(β - α))/(cos β cos α)
(x from the given question) = (h tan β)/(tan β - tan α)
Therefore, the height of the tower is [(h tan β)/(tan β - tan α)], which is option 'c'.
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From the top and the foot of a house h metres high the angles of elevation of the top of a tower are β and α respectively, then the height of the tower isa)[(h sinβ)/(cosβ-sinα)]b)[(h cosβ)/(cosβ-cosα)]c)[(h tanβ)/(tanβ-tanα)]d)[(h cotβ)/(cotβ-cotα)]Correct answer is option 'C'. Can you explain this answer?
Question Description
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