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Question

A vertical lamp-post of height 9 meters stands at the corner of a rectangular field.The angle of elevation of its top from the farthest corner is $$\displaystyle 30^{\circ}$$,while from another corner it is $$\displaystyle 45^{\circ}.$$ The area of the field is 


A
92meter2
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B
813meter2
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C
812meter2
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D
93meter2
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Solution

The correct option is D $$\displaystyle 81\sqrt{2}\: meter\:^{2}$$
Consider rectangle $$ABCD$$ be the rectangular field. Let $$PD$$ be the height of vertical lamp post.

it is given that Height of the Pole $$=PD=9$$m

Now in $$\triangle PDA$$
$$tan45^o=\dfrac{PD}{DA}=\dfrac{9}{DA}\Rightarrow DA=9$$

Now in $$\triangle PDB$$
$$tan30^o=\dfrac{9}{DB}\Rightarrow DB=9\sqrt3$$

Also, we can see that $$\triangle DAB$$ is Right Angled Triangle
$$\therefore DA^2+AB^2=DB^2\Rightarrow 9^2+AB^2=(9\sqrt3)^2\Rightarrow AB^2=81\times2\Rightarrow AB=9\sqrt2$$

Now, Area of Rectangle $$=DA\times AB=9\sqrt2\times 9=81\sqrt2$$ $$m^2$$
 
Therefore, option (C) is the correct answer.

879094_134889_ans_67f27c182e6548459258535a75182962.png

Mathematics

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