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If the angles of elevation of the top of...

If the angles of elevation of the top of a tower from two points distance s and `t(s gt t)` from its foot are 30° and 60° respectively, then the height of the tower is

A

`sqrt(s+t)`

B

`sqrt(st)`

C

`sqrt(s-t)`

D

`sqrt(s/t)`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we will use the information provided about the angles of elevation and the distances from the tower. ### Step 1: Understand the Problem We have a tower and two points A and B from which the angles of elevation to the top of the tower are given. The distances from the foot of the tower to points A and B are `s` and `t` respectively, with `s > t`. The angles of elevation from points A and B are 30° and 60° respectively. ### Step 2: Set Up the Right Triangles From point A (distance `s` from the tower), the angle of elevation is 30°. From point B (distance `t` from the tower), the angle of elevation is 60°. ### Step 3: Write the Tangent Equations Using the definition of tangent in right triangles, we can write: 1. For point A: \[ \tan(30^\circ) = \frac{h}{s} \] We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), so: \[ \frac{1}{\sqrt{3}} = \frac{h}{s} \] Rearranging gives: \[ h = \frac{s}{\sqrt{3}} \quad \text{(Equation 1)} \] 2. For point B: \[ \tan(60^\circ) = \frac{h}{t} \] We know that \(\tan(60^\circ) = \sqrt{3}\), so: \[ \sqrt{3} = \frac{h}{t} \] Rearranging gives: \[ h = \sqrt{3} \cdot t \quad \text{(Equation 2)} \] ### Step 4: Equate the Two Expressions for Height From Equation 1 and Equation 2, we have two expressions for the height \(h\): 1. \(h = \frac{s}{\sqrt{3}}\) 2. \(h = \sqrt{3} \cdot t\) Setting these equal to each other: \[ \frac{s}{\sqrt{3}} = \sqrt{3} \cdot t \] ### Step 5: Solve for Height Cross-multiplying gives: \[ s = 3t \] Now, substituting \(t = \frac{s}{3}\) back into either expression for \(h\): Using \(h = \sqrt{3} \cdot t\): \[ h = \sqrt{3} \cdot \frac{s}{3} = \frac{s \sqrt{3}}{3} \] ### Step 6: Final Expression for Height To express \(h\) in terms of both \(s\) and \(t\), we can use the relationship \(t = \frac{s}{3}\): \[ h = \sqrt{s \cdot t} \] ### Conclusion Thus, the height of the tower \(h\) is given by: \[ h = \sqrt{s \cdot t} \]
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MCGROW HILL PUBLICATION-HEIGHTS AND DISTANCES -MULTIPLE CHOICE QUESTIONS
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  14. A tree is broken by wind, its upper part touches the ground at a point...

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  15. From the top of a light house, the angles of depression of two station...

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  17. An aeroplane flying horizontally 900 m above the ground is observed at...

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  18. From the top of the house 18m high, if the angle of elevation of the t...

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  19. If the angle of elevation of an object from a point 100 m above a lake...

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