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A window of a house is h metre above the ground. From the window, the angles of elevation and depression of the top and bottom of another house situated on the opposite side of the lane are found to be alpha and beta respectively.Prove that the height of the house is h(1 tanalpha tanbeta) metres.?
Verified Answer
A window of a house is h metre above the ground. From the window, the ...
height of another house is h(1 + tanα.cotβ)
see the attachment,
Let AE is the window of one house which is h m metre above from the ground . from the window , angle of elevation (α ) and angle of depression ( β ) of the top and bottom of another house of height BD is shown in figure.
now, AE = BC = h
from ∆BCE ,
tanβ = BC/CE = h/CE
CE = h/tanβ
from ∆DEC ,
tanα = DC/CE
from equation (1) CE = h/tanβ = hcotβ, put it here,
tanα = DC/hcotβ
DC = h.tanα.cotβ
now, we can see that height of another house is BD
BD = BC + CD
from equations (1) and (3)
BD = h + h.tanα.cotβ = h(1 + tanα.cotβ)
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Most Upvoted Answer
A window of a house is h metre above the ground. From the window, the ...
Understanding the Problem
In this scenario, we analyze a window at height h metres above the ground. The angles of elevation (alpha) and depression (beta) are observed from this window to the top and bottom of an opposite house, respectively.
Visual Representation
- Imagine a vertical line representing the height of the opposite house.
- The window is at a height of h metres.
- From the window, draw angles alpha (upwards) to the top of the house and beta (downwards) to the bottom.
Using Trigonometry
1. Top of the Opposite House:
- The distance from the window to the top of the opposite house can be expressed using the tangent function:
- Height from window to top = (height of opposite house - h) = d * tan(alpha)
2. Bottom of the Opposite House:
- The distance from the window to the bottom can similarly be expressed:
- Height from window to bottom = (h - 0) = d * tan(beta)
Setting up the Equation
- Let the height of the opposite house be H.
- Then, we can write two equations based on the above observations:
- H - h = d * tan(alpha)
- h = d * tan(beta)
Solving for Height
- Rearranging these equations gives us:
- H = h + d * tan(alpha)
- d = h / tan(beta)
- Substituting d into the equation for H:
- H = h + (h / tan(beta)) * tan(alpha)
Final Result
- Simplifying gives:
- H = h(1 + tan(alpha) * tan(beta))
Thus, the height of the opposite house is h(1 + tan(alpha) * tan(beta)) metres.
In this scenario, we analyze a window at height h metres above the ground. The angles of elevation (alpha) and depression (beta) are observed from this window to the top and bottom of an opposite house, respectively.
Visual Representation
- Imagine a vertical line representing the height of the opposite house.
- The window is at a height of h metres.
- From the window, draw angles alpha (upwards) to the top of the house and beta (downwards) to the bottom.
Using Trigonometry
1. Top of the Opposite House:
- The distance from the window to the top of the opposite house can be expressed using the tangent function:
- Height from window to top = (height of opposite house - h) = d * tan(alpha)
2. Bottom of the Opposite House:
- The distance from the window to the bottom can similarly be expressed:
- Height from window to bottom = (h - 0) = d * tan(beta)
Setting up the Equation
- Let the height of the opposite house be H.
- Then, we can write two equations based on the above observations:
- H - h = d * tan(alpha)
- h = d * tan(beta)
Solving for Height
- Rearranging these equations gives us:
- H = h + d * tan(alpha)
- d = h / tan(beta)
- Substituting d into the equation for H:
- H = h + (h / tan(beta)) * tan(alpha)
Final Result
- Simplifying gives:
- H = h(1 + tan(alpha) * tan(beta))
Thus, the height of the opposite house is h(1 + tan(alpha) * tan(beta)) metres.
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A window of a house is h metre above the ground. From the window, the angles of elevation and depression of the top and bottom of another house situated on the opposite side of the lane are found to be alpha and beta respectively.Prove that the height of the house is h(1 tanalpha tanbeta) metres.?
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A window of a house is h metre above the ground. From the window, the angles of elevation and depression of the top and bottom of another house situated on the opposite side of the lane are found to be alpha and beta respectively.Prove that the height of the house is h(1 tanalpha tanbeta) metres.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about A window of a house is h metre above the ground. From the window, the angles of elevation and depression of the top and bottom of another house situated on the opposite side of the lane are found to be alpha and beta respectively.Prove that the height of the house is h(1 tanalpha tanbeta) metres.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A window of a house is h metre above the ground. From the window, the angles of elevation and depression of the top and bottom of another house situated on the opposite side of the lane are found to be alpha and beta respectively.Prove that the height of the house is h(1 tanalpha tanbeta) metres.?.
A window of a house is h metre above the ground. From the window, the angles of elevation and depression of the top and bottom of another house situated on the opposite side of the lane are found to be alpha and beta respectively.Prove that the height of the house is h(1 tanalpha tanbeta) metres.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about A window of a house is h metre above the ground. From the window, the angles of elevation and depression of the top and bottom of another house situated on the opposite side of the lane are found to be alpha and beta respectively.Prove that the height of the house is h(1 tanalpha tanbeta) metres.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A window of a house is h metre above the ground. From the window, the angles of elevation and depression of the top and bottom of another house situated on the opposite side of the lane are found to be alpha and beta respectively.Prove that the height of the house is h(1 tanalpha tanbeta) metres.?.
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