Question

A vertical pole of height 10 metres stands at one corner of a rectangular field. The angle of elevation of its top from the farthest corner is 30°, while that from another corner is 60°. The area (in m^2)of rectangular
field is​

Answer 1

Answer

Area of rectangular field is $\sf{\frac{200\sqrt{2}}{3}\:m^2}$

$\rule{100}2$

Explanation

[ Refer the attachment for figure ]

Assume a rectangular field of sides ABCD such that AB = CD = x and AD = BC = y.

A vertical pole (ED) of height 10 m stands at one corner of a rectangular field.

The angle of elevation from the top and bottom of the tower at the bottom or end of the rectangular field is 30° or from the farthest corner is 30°.

And angle of elevation from the top of tower to the top of rectangular field is 60°.

In ∆DAB

By Pythagoras Theoram

$\implies\:\sf{H^2\:=\:P^2\:+\:B^2}$

$\implies\:\sf{(BD)^2\:=\:(AB)^2\:+\:(AD)^2}$

$\implies\:\sf{(BD)^2\:=\:x^2\:+\:y^2}$

$\implies\:\sf{BD\:=\:\sqrt{x^2\:+\:y^2}}$ ---- [1]

In ∆EDC

$\implies\:\sf{tan60^{\circ}\:=\:\dfrac{P}{B}}$

$\implies\:\sf{tan60^{\circ}\:=\:\dfrac{ED}{DC}}$

$\implies\:\sf{\sqrt{3}\:=\:\dfrac{10}{x}}$

$\implies\:\sf{x\:=\:\dfrac{10}{\sqrt{3}}}$ ---- [2]

In ∆EDB

$\implies\:\sf{tan30^{\circ}\:=\:\dfrac{ED}{BD}}$

$\implies\:\sf{\dfrac{1}{\sqrt{3}}\:=\:\dfrac{10}{BD}}$

Cross-multiply them

$\implies\:\sf{BD\:=\:10\sqrt{3}}$

$\implies\:\sf{\sqrt{x^2\:+\:y^2}\:=\:10\sqrt{3}}$ [From (1)]

$\implies\:\sf{x^2\:+\:y^2\:=\:(10\sqrt{3})^2}$

$\implies\:\sf{x^2\:+\:y^2\:=\:300}$ ---- [3]

Substitute value of x = 10/√3 in equation (3)

$\implies\:\sf{\bigg(\dfrac{10}{\sqrt{3}}\bigg)^2\:+\:y^2\:=\:300}$

$\implies\:\sf{\dfrac{100}{3}\:+\:y^2\:=\:300}$

$\implies\:\sf{\dfrac{100\:+\:3y^2}{3}\:=\:300}$

$\implies\:\sf{100\:+\:3y^2\:=\:900}$

$\implies\:\sf{3y^2\:=\:800}$

$\implies\:\sf{y^2\:=\:\dfrac{800}{3}}$

$\implies \: \sf y \: = \: \dfrac{20 \sqrt{2} }{ \sqrt{3} }$

We have to find the area of the rectangular field.

From above calculations we have -

• x = length = 10/√3 m
• y = breadth = 20√2/√3 m

Area of rectangle = length × breadth

$\implies\:\sf{x\:\times\:y}$

$\implies\:\sf{\dfrac{10}{\sqrt{3}}\:\times\:\dfrac{20\sqrt{2}}{\sqrt{3}}}$

We know that √3 × √3 = (√3)² = 3

$\implies\:\sf{\dfrac{200\sqrt{2}}{3}}$

Answer 2

${\large\bf{\mid{\overline{\underline{Given:-}}}\mid}}$

$\red{\textbf{Refer To image First}} \:$

• Height of Pole = 10m = AE
• Angle of Elevation of Top of Pole From Farthest point = angle ECA = 30°
• Angle of Elevation of Another corner = angle EDA = 60°

$\large\star{\underline{\tt{\red{Answer}}}}\star$

In ∆ EDA ,

→ angle EDA = 60°

→ EA = Pole = 10m

Hence,

$tan60 = \frac{p}{b} = \frac{EA }{DA} \\ \\ \red\leadsto \: \sqrt{3} = \frac{10}{DA} \\ \\ DA \: = \frac{10}{ \sqrt{3} } = CB = l \: (length)$

__________________________

Now,

In ∆EAC,

→ angle ECA = 30°

→ EA = 10m = Pole

→ CA = Diagonal of Rectangle ABCD =√( l² + b² )

Hence ,,

$tan30 = \frac{EA}{ CA} \\ \\ \red\leadsto \: \frac{1}{ \sqrt{3} } = \frac{10}{ \sqrt{ {l}^{2} + {b}^{2}} } \\ \\ \red\leadsto \: \large\boxed{\bold{ \sqrt{ {l}^{2} + {b}^{2}} = 10 \sqrt{3} }} \\ \\ squaring \: both \: sides \: we \: get \\ \\ \red\leadsto \: {l}^{2} + {b}^{2} = 300$

___________________________

Now, putting value of CB = length of Rectangle we get,,

$( \frac{10}{ \sqrt{3} } )^{2} + {b}^{2} = 300 \\ \\ \red{\boxed\implies} \: \: {b}^{2} = 300 - \frac{100}{3} \\ \\ \red{\boxed\implies} \: \: \: {b}^{2} = \frac{800}{3} \\ \\ \red{\boxed\implies} \: \: b = \frac{20 \sqrt{2} }{ \sqrt{3} }$

Now,

we know that,,,

$\large\boxed{\bold{Area \: of \: Rectangle = l×b}} \: \\ \\ putting \: values \: \\ \\ \red\leadsto \: Area \: = \frac{10}{ \sqrt{3} } \times \frac{20 \sqrt{2} }{ \sqrt{3} } \\ \\ \red\leadsto \: \pink{\large\boxed{\boxed{\bold{Area = \frac{200 \sqrt{2} }{3} {m}^{2} }}}}$

Hence, Area of Rectangle ABCD will be (200√2/3)m² ..

______________________________

$\large\bold\star\underline\mathcal{Extra\:Brainly\:Knowledge:-}$

1) Each of the interior angles of a rectangle is 90°.

2) The diagonals of a rectangle bisect each other.

3) The opposite sides of a rectangle are parallel.

4) The opposite sides of a rectangle are equal.

5) A rectangle whose side lengths are a and b has area = a×b×sin90° = a×b

6) A rectangle whose side lengths are a a and b b has perimeter 2(a + b)...

7) The length of each diagonal of a rectangle whose side lengths are a and b is √(a²+b²) ..