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If the angle of elevation of a tower from two distant points a and b (a > b) from its foot and In the same straight line and on the same side of it are \( \Large 30 ^{\circ} \) and \( \Large 60 ^{\circ} \), then the height of the tower is

A) \( \Large \sqrt{\frac{a}{b}} \)
B) \( \Large \sqrt{a+b} \)
C) \( \Large \sqrt{ab} \)
D) \( \Large \sqrt{a-b} \)

Correct Answer:



C) \( \Large \sqrt{ab} \)

Description for Correct answer:

Let AB be the tower.

Where, AB = h m

In \( \Large \triangle ABC, \tan 60 ^{\circ} =\frac{h}{b} \) ...(i)

In \( \Large \triangle ABD, \tan 30 ^{\circ} =\frac{h}{a} \) ...(ii)

On multiplying Eqs. (i) and (ii), we get

\( \Large \tan 60 ^{\circ} \tan 30 ^{\circ} = \frac{h^{2}}{ab} \)

=> \( \Large \sqrt{3} \times \frac{1}{\sqrt{3}}=\frac{h^{2}}{ab} => h^{2} = ab \)

\( \Large \therefore h = \sqrt{ab} \)

Part of solved Height and Distance questions and answers : >> Elementary Mathematics >> Height and Distance

Comments

Similar Questions

1). The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is \( \Large 60 ^{\circ} \), and the angle of elevation of the top of the second tower from the foot of the first tower is \( \Large 30 ^{\circ} \). The distance between the two towers is n times the height of the shorter tower. What is n equal to?

A). \( \Large \sqrt{2} \)

B). \( \Large \sqrt{3} \)

C). \( \Large \frac{1}{2} \)

D). \( \Large \frac{1}{3} \)

2). At the foot of a mountain, the elevation of its summit is \( \Large 45 ^{\circ} \). After ascending 2km towards the mountain upon an incline of \( \Large 30 ^{\circ} \), the elevation changes to \( \Large 60 ^{\circ} \). The height of the mountain is

A). \( \Large \left(\sqrt{3} - 1\right) km \)

B). \( \Large \left(\sqrt{3} + 1\right) km \)

C). \( \Large \left(\sqrt{3} + 2\right) km \)

D). \( \Large \left(\sqrt{3} - 2\right) km \)

3). A man standing in one corner of a square football field observes that the angle subtended by a pole in the corner just diagonally opposite to this corner is \( \Large 60 ^{\circ} \). When he retires 80 m from the corner, along the same straight line, he finds the angle to be \( \Large 30 ^{\circ} \). The length of the field is

A). 20 m

B). \( \Large 40\sqrt{2} \) m

C). 40 m

D). \( \Large 20\sqrt{2} \) m

4). A spherical balloon of radius r subtends angle \( \Large 60 ^{\circ} \) at the eye of an observer. If the angle of elevation of its centre is \( \Large 60 ^{\circ} \) and h is the height of the centre of the balloon, then which one of the following is correct?

A). h=r

B). \( \Large h = \sqrt{2}r \)

C). \( \Large h = \sqrt{3}r \)

D). h=2r

5). If the roots of equations \( \Large ax^{2}+bx+c=0 \) be \( \Large \ \alpha \ and\ \beta \) then the roots of equations \( \Large cx^{2}+bx+a=0 \) are:

A). \( \Large - \alpha ,\ - \beta \)

B). \( \Large \alpha ,\ \frac{1}{ \beta } \)

C). \( \Large \frac{1}{ \alpha },\ \frac{1}{ \beta } \)

D). none of these

6). Both the roots of the given equation. \( \Large \left(x-a\right) \left(x-b\right)+ \left(x-b\right) \left(x-c\right)+ \left(x-c\right) \left(x-a\right)=0 \) are always

A). positive

B). negative

C). real

D). imaginary

7). If the difference of roots of the equation \( \Large x^{2}-bx+c=0 \) be 1, then;

A). \( \Large b^{2}-4c-1=0 \)

B). \( \Large b^{2}-4c=0 \)

C). \( \Large b^{2}-4c+1=0 \)

D). \( \Large b^{2}+4c-1=0 \)

8). If \( \Large 2+i\sqrt{3} \) is a root of the equation \( \Large x^{2}+px+q=0 \) where p and q are real, then \( \Large p, q \) is equal to:

A). (-4, 7)

B). (4,-7)

C). (4, 7)

D). (-4, -7)

9). If \( \Large x=\sqrt{1+\sqrt{1+\sqrt{1}+.....}} \), the x is equal to

A). \( \Large \frac{1+\sqrt{5}}{2} \)

B). \( \Large \frac{1-\sqrt{5}}{2} \)

C). \( \Large \frac{1\pm \sqrt{5}}{2} \)

D). none of these .

10). If \( \Large \alpha , \beta \) are the roots of equation \( \Large ax^{2}+bx+c=0 \), then \( \Large \frac{ \alpha }{ \alpha \beta +b}+\frac{ \beta }{ \alpha x+b} \) is equal to

A). 2/a

B). 2/b

C). 2/c

D). -2/a