The horizontal distance between two towers is 90 m and the angular depression of the top of the first as seen from the top of the second is 30o. Find the difference of the heights of the two towers.a)30 meters.b)30√4meters.c)30√3meters.d)90 meters.Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Solution:

Given:
Distance between the two towers (horizontal distance) = 90 m
Angular depression of the top of the first tower as seen from the top of the second tower = 30 degrees

To find:
Difference in heights of the two towers

Let's assume the height of the first tower is h1 and the height of the second tower is h2.

We can form a right-angled triangle by considering the first tower, the second tower, and the line connecting the top of the first tower to the top of the second tower.

Using trigonometry, we can relate the angles and sides of a right-angled triangle.

1. Identify the key components:
- Distance between the two towers (horizontal distance) = 90 m
- Angular depression of the top of the first tower as seen from the top of the second tower = 30 degrees
- Height of the first tower = h1
- Height of the second tower = h2
- Difference in heights of the two towers = h2 - h1 (to be determined)

2. Form the right-angled triangle:
- The horizontal distance between the two towers forms the base of the triangle.
- The difference in heights of the two towers forms the vertical side of the triangle.
- The line connecting the top of the first tower to the top of the second tower forms the hypotenuse of the triangle.

3. Apply trigonometry:
- The tangent of the angle of depression is equal to the ratio of the opposite side to the adjacent side.
- In this case, the opposite side is the difference in heights (h2 - h1) and the adjacent side is the horizontal distance (90 m).

Therefore, we have:
tan(30 degrees) = (h2 - h1) / 90

4. Solve the equation:
Using the value of tan(30 degrees) = 1/√3, we can solve for (h2 - h1):
1/√3 = (h2 - h1) / 90

Cross-multiplying the equation:
h2 - h1 = 90 / √3

Rationalizing the denominator by multiplying both the numerator and denominator by √3:
h2 - h1 = 90√3 / 3

Simplifying the expression:
h2 - h1 = 30√3

5. Final answer:
Therefore, the difference in heights of the two towers is 30√3 meters.

Hence, the correct answer is option 'C' - 30√3 meters.

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