The horizontal distance between two towers is 40√3 m. The angle of depression of the top of the first tower when seen from the top of the second tower is 30°. If the height of the second tower is 130 m, find the height of the first tower.
90 m
Solving Tower Height Problems Using Trigonometry
This problem involves finding the height of a tower using the angle of depression and the horizontal distance between two towers. We can solve this by applying basic trigonometry principles, specifically the tangent function, within a right-angled triangle formed by the towers and the horizontal distance.
Understanding the Setup
Let's visualize the scenario:
- We have two towers. Let's call them Tower 1 and Tower 2.
- The horizontal distance between their bases is given as $40\sqrt{3}$ meters.
- The height of the second tower (Tower 2) is 130 meters.
- The angle of depression from the top of the second tower to the top of the first tower (Tower 1) is 30°.
The angle of depression is the angle between the horizontal line of sight from the observer (at the top of Tower 2) and the line of sight downwards to an object (the top of Tower 1). This angle is equal to the angle of elevation from the top of Tower 1 to the top of Tower 2 (alternate interior angles).
Formulating the Trigonometric Relationship
Let:
- $H_1$ be the height of the first tower.
- $H_2$ be the height of the second tower ($H_2 = 130$ m).
- $D$ be the horizontal distance between the towers ($D = 40\sqrt{3}$ m).
- $\theta$ be the angle of depression from the top of Tower 2 to the top of Tower 1 ($\theta = 30°$).
Imagine a horizontal line drawn from the top of Tower 2. The point where this line meets the vertical line extended upwards from the top of Tower 1 forms a right-angled triangle. The vertical side of this triangle represents the difference in height between the two towers, $H_2 - H_1$. The horizontal side is the distance between the towers, $D$. The angle opposite the vertical side is the angle of depression, $\theta = 30°$.
In this right-angled triangle, we can use the tangent function:
\(\tan(\theta) = \frac{\text{Opposite side}}{\text{Adjacent side}}\)
\(\tan(\theta) = \frac{H_2 - H_1}{D}\)
Calculating the Height of the First Tower
We are given $\theta = 30°$, $H_2 = 130$ m, and $D = 40\sqrt{3}$ m. We know that $\tan(30°) = \frac{1}{\sqrt{3}}$.
Substitute these values into the equation:
\(\tan(30°) = \frac{130 - H_1}{40\sqrt{3}}\)
\(\frac{1}{\sqrt{3}} = \frac{130 - H_1}{40\sqrt{3}}\)
Now, we solve for $H_1$:
Multiply both sides by $40\sqrt{3}$:
\(\frac{1}{\sqrt{3}} \times 40\sqrt{3} = 130 - H_1\)
\(40 = 130 - H_1\)
Rearrange the equation to find $H_1$:
\(H_1 = 130 - 40\)
\(H_1 = 90\)
The height of the first tower is 90 meters.
Summary of Calculation Steps
- Identify the known quantities: Horizontal distance, height of Tower 2, angle of depression.
- Draw a diagram representing the scenario, showing the towers, distance, and the right triangle formed by the difference in heights and the horizontal distance.
- Use the tangent function relating the angle of depression to the difference in heights and the horizontal distance: $\tan(\theta) = \frac{H_2 - H_1}{D}$.
- Substitute the given values into the equation.
- Solve the equation for the unknown height, $H_1$.
| Parameter | Value |
|---|---|
| Horizontal Distance (D) | $40\sqrt{3}$ m |
| Height of Second Tower ($H_2$) | 130 m |
| Angle of Depression ($\theta$) | 30° |
| Value of $\tan(30°)$ | $\frac{1}{\sqrt{3}}$ |
| Height of First Tower ($H_1$) | ? |
The calculated height of the first tower is 90 meters.
Revision Table: Key Concepts
| Concept | Explanation |
|---|---|
| Angle of Depression | The angle between the horizontal line from the observer's eye and the line of sight to an object below the horizontal. |
| Angle of Elevation | The angle between the horizontal line from the observer's eye and the line of sight to an object above the horizontal. (Note: Angle of depression from A to B equals angle of elevation from B to A). |
| Trigonometric Ratios | Relationships between the angles and sides of a right-angled triangle (Sine, Cosine, Tangent). |
| Tangent ($\tan$) | In a right-angled triangle, $\tan(\text{angle}) = \frac{\text{Opposite side}}{\text{Adjacent side}}$. |
Additional Information: Angles and Heights
Problems involving angles of elevation and depression are common applications of trigonometry. They typically involve setting up a right-angled triangle based on the given information and using the appropriate trigonometric ratio (sin, cos, or tan) to find an unknown side or angle.
- Always draw a clear diagram to represent the situation.
- Identify the right triangle(s) in the diagram.
- Determine which angle is given or needs to be found (angle of elevation or depression).
- Identify the known and unknown sides relative to the angle (opposite, adjacent, hypotenuse).
- Choose the correct trigonometric ratio that relates the known and unknown sides to the angle.
- Solve the resulting equation.
Remember that the angle of depression is measured downwards from a horizontal line, and the angle of elevation is measured upwards from a horizontal line. These horizontal lines are crucial for defining the angles correctly.
Important Questions from Heights and Distances
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