From a height of h units, a man observes the angle of elevation as α and angle of depression as β of the top and bottom respectively of a tower of height H (> 4h). To what further height should be climb so that the values of angle of elevation and angle of depression get interchanged for the top and bottom of the tower?
H - 2h units
Understanding Angles of Elevation and Depression
This problem involves trigonometry, specifically the concepts of the angle of elevation and the angle of depression. The angle of elevation is the angle measured upwards from a horizontal line to a point above the observer. The angle of depression is the angle measured downwards from a horizontal line to a point below the observer.
Initial Observation Setup
Let's visualize the initial situation. The man is at a height \(h\). He observes a tower of height \(H\). Let the horizontal distance from the man to the tower be \(x\).
- The top of the tower is observed with an angle of elevation \(\alpha\). This means the angle between the horizontal line from the man's eye level to the tower and the line of sight to the top of the tower is \(\alpha\). The height of the tower above the man's eye level is \(H - h\).
- The bottom of the tower is observed with an angle of depression \(\beta\). This means the angle between the horizontal line from the man's eye level to the tower and the line of sight to the bottom of the tower is \(\beta\). By alternate angles, the angle formed at the base of the tower between the horizontal ground and the line of sight from the man is also \(\beta\). The vertical distance from the man's eye level to the base of the tower is \(h\).
Setting up Equations from Initial Observation
Using trigonometry (tangent function) in the right-angled triangles formed:
- For the angle of elevation \(\alpha\) to the top of the tower: The opposite side is the height of the tower above the man's level (\(H - h\)), and the adjacent side is the horizontal distance \(x\). \( \tan \alpha = \frac{H - h}{x} \) This gives us \(x = \frac{H - h}{\tan \alpha}\).
- For the angle of depression \(\beta\) to the bottom of the tower: The opposite side is the man's height above the ground (\(h\)), and the adjacent side is the horizontal distance \(x\). \( \tan \beta = \frac{h}{x} \) This gives us \(x = \frac{h}{\tan \beta}\).
Equating the two expressions for \(x\):
\( \frac{H - h}{\tan \alpha} = \frac{h}{\tan \beta} \) \( (H - h) \tan \beta = h \tan \alpha \quad \quad (1) \)New Observation Setup After Climbing
Now, the man climbs further to a new height, let's call it \(h'\). The additional height climbed is \(h' - h\). From this new height \(h'\), the angles of elevation and depression are interchanged.
- The angle of elevation to the top of the tower is now \(\beta\). The height of the tower above the man's new eye level is \(H - h'\). The horizontal distance is still \(x\).
- The angle of depression to the bottom of the tower is now \(\alpha\). The vertical distance from the man's new eye level to the base of the tower is \(h'\). The horizontal distance is still \(x\).
Setting up Equations from New Observation
Using trigonometry again:
- For the new angle of elevation \(\beta\) to the top of the tower: \( \tan \beta = \frac{H - h'}{x} \) This gives us \(x = \frac{H - h'}{\tan \beta}\).
- For the new angle of depression \(\alpha\) to the bottom of the tower: \( \tan \alpha = \frac{h'}{x} \) This gives us \(x = \frac{h'}{\tan \alpha}\).
Equating the new expressions for \(x\):
\( \frac{H - h'}{\tan \beta} = \frac{h'}{\tan \alpha} \) \( (H - h') \tan \alpha = h' \tan \beta \quad \quad (2) \)Solving for the New Height \(h'\)
We have two equations:
\( (H - h) \tan \beta = h \tan \alpha \quad \quad (1) \) \( (H - h') \tan \alpha = h' \tan \beta \quad \quad (2) \)From equation (1), we can write:
\( \frac{\tan \alpha}{\tan \beta} = \frac{H - h}{h} \)From equation (2), we can write:
\( \frac{\tan \alpha}{\tan \beta} = \frac{h'}{H - h'} \)Equating the expressions for \(\frac{\tan \alpha}{\tan \beta}\):
\( \frac{H - h}{h} = \frac{h'}{H - h'} \)Cross-multiplying:
\( (H - h)(H - h') = h h' \) \( H^2 - Hh' - Hh + hh' = hh' \) \( H^2 - Hh = Hh' \) \( H(H - h) = Hh' \)Since \(H\) is the height of the tower and \(H > 4h\), \(H \neq 0\). We can divide both sides by \(H\):
\( H - h = h' \)So, the new height the man should climb to is \(h' = H - h\).
Calculating the Additional Height
The question asks for the *further* height the man should climb. This is the difference between the new height \(h'\) and the original height \(h\).
Additional height \(= h' - h\)
Substitute the value of \(h'\):
Additional height \(= (H - h) - h = H - 2h\).
Therefore, the man should climb a further \(H - 2h\) units.
Summary of the Calculation
| Initial Setup (Height \(h\)) | New Setup (Height \(h'\)) |
|---|---|
| Angle of elevation to top: \(\alpha\) | Angle of elevation to top: \(\beta\) |
| Angle of depression to bottom: \(\beta\) | Angle of depression to bottom: \(\alpha\) |
| \(\tan \alpha = \frac{H-h}{x}\) | \(\tan \beta = \frac{H-h'}{x}\) |
| \(\tan \beta = \frac{h}{x}\) | \(\tan \alpha = \frac{h'}{x}\) |
| \(\frac{\tan \alpha}{\tan \beta} = \frac{H-h}{h}\) | \(\frac{\tan \alpha}{\tan \beta} = \frac{h'}{H-h'}\) |
| Equating ratios: \(\frac{H-h}{h} = \frac{h'}{H-h'}\) | |
| Solving for \(h'\): \(h' = H - h\) | |
| Additional height: \(h' - h = (H - h) - h = H - 2h\) | |
Revision Table: Angles of Elevation and Depression
| Concept | Definition | Diagramatic Representation |
|---|---|---|
| Angle of Elevation | Angle measured upwards from the horizontal line of sight to an object above the observer. | Horizontal line \(\to\) Object above observer. Angle is between horizontal and line of sight. |
| Angle of Depression | Angle measured downwards from the horizontal line of sight to an object below the observer. | Horizontal line \(\to\) Object below observer. Angle is between horizontal and line of sight. (Alternate interior angle with angle at object level is often useful). |
Additional Information: Trigonometry in Heights and Distances
Problems involving heights and distances often use basic trigonometric ratios (sine, cosine, tangent) applied to right-angled triangles. Key steps usually involve:
- Drawing a clear diagram representing the situation.
- Identifying the right-angled triangles involved.
- Labeling known heights, distances, and angles.
- Assigning variables to unknown quantities.
- Using trigonometric ratios (\(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\), \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\)) to set up equations.
- Solving the system of equations to find the unknown values.
- Remembering that the angle of depression from point A to point B is equal to the angle of elevation from point B to point A, assuming they are at different vertical levels.
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