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
B
813meter2
C
812meter2
D
93meter2

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 $$=PD=9$$mNow 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.Mathematics

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