A vertical pole and a vertical tower are on the same level ground in such a way that, from the top of the pole, the angle of elevation of the top of the tower is 60° and the angle of depression of the bottom of the tower is 30°. If the height of the pole is 24 m, then find the height of the tower (in m).
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Calculating Tower Height Using Angle of Elevation and Depression
This problem involves using trigonometry, specifically the tangent function, to find the height of a tower given the height of a pole and the angles of elevation and depression from the top of the pole to the top and bottom of the tower, respectively.
Understanding the Geometry
Let's visualize the scenario:
- Let PQ be the vertical pole with height 24 m. P is the top, and Q is the bottom.
- Let TB be the vertical tower. T is the top, and B is the bottom.
- The pole and tower are on the same level ground, so Q and B are on the ground.
- Draw a horizontal line from the top of the pole P, meeting the tower at point R.
- The angle of elevation of the top of the tower (T) from P is ∠TPR = 60°.
- The angle of depression of the bottom of the tower (B) from P is ∠RPB = 30°.
- The figure PQBR forms a rectangle, where PQ is parallel to RB and PR is parallel to QB.
- Therefore, PQ = RB = 24 m and PR = QB. PR is the horizontal distance between the pole and the tower.
- The height of the tower is TB = TR + RB.
Applying Trigonometry to Find Distances
We can use the tangent function ($\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$) in the right triangles formed.
Step 1: Find the horizontal distance PR (or QB)
Consider the right triangle ▵PBR. The angle of depression from P to B is 30°. In this triangle, RB is the side opposite to the angle ∠RPB (which is 30°), and PR is the side adjacent to this angle. We know RB = 24 m (height of the pole).
Using tangent:
We know that $\tan(30^\circ) = \frac{1}{\sqrt3}$.
Solving for PR:
So, the horizontal distance between the pole and the tower is $24\sqrt3$ m.
Step 2: Find the height TR
Now consider the right triangle ▵PTR. The angle of elevation from P to T is 60°. In this triangle, TR is the side opposite to ∠TPR (which is 60°), and PR is the side adjacent to this angle. We just found PR = $24\sqrt3$ m.
Using tangent:
We know that $\tan(60^\circ) = \sqrt3$.
Solving for TR:
Step 3: Calculate the total height of the tower
The total height of the tower TB is the sum of TR and RB.
Summary of Calculations
| Triangle | Angle | Opposite Side | Adjacent Side | Tangent Equation | Calculated Value |
|---|---|---|---|---|---|
| ▵PBR | ∠RPB = 30° | RB = 24 m | PR | $\tan(30^\circ) = \frac{24}{PR}$ | PR = $24\sqrt3$ m |
| ▵PTR | ∠TPR = 60° | TR | PR = $24\sqrt3$ m | $\tan(60^\circ) = \frac{TR}{24\sqrt3}$ | TR = 72 m |
Total Tower Height = TR + RB = 72 m + 24 m = 96 m.
Thus, the height of the tower is 96 m.
Revision Table: Key Concepts
| Concept | Description | Relevance to Problem |
|---|---|---|
| Angle of Elevation | The angle between the horizontal line of sight and the line of sight upwards to an object. | Used to find the upper part of the tower's height (TR). |
| Angle of Depression | The angle between the horizontal line of sight and the line of sight downwards to an object. | Used to find the horizontal distance between the pole and tower (PR) using the pole's height (RB). |
| Trigonometric Ratios | Relationships between angles and sides of right triangles (SOH CAH TOA). | Tangent ($\tan$) is used as it relates the opposite and adjacent sides. |
| Right Triangle | A triangle with one angle equal to 90 degrees. | ▵PBR and ▵PTR are right triangles formed by the horizontal line and vertical objects. |
Additional Information: Angles and Heights
Problems involving angles of elevation and depression often require drawing a clear diagram and identifying the right triangles. The horizontal line of sight from the observer's position is crucial for defining these angles correctly. The angle of depression from point P to point B is equal to the angle of elevation from point B to point P (alternate interior angles if the horizontal line is parallel to the ground).
In this specific problem:
- The height of the pole gives us the height of the rectangle part below the horizontal line from the top of the pole.
- The angle of depression helps find the horizontal distance using the pole's height.
- The angle of elevation helps find the remaining height of the tower above the horizontal line using the horizontal distance.
- The total height of the taller object (tower) is the sum of the height equal to the shorter object (pole) and the height calculated using the angle of elevation.
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