Question

A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8m. find the height of the tree.

please answer the question​

Answer 1

$\underline{\underline{\sf{\maltese\:\:Question}}}$

• A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8m. Find the Height  of the Tree.

$\underline{\underline{\sf{\maltese\:\:Given}}}$

• The distance between foot of the tree to point where the top touches the ground = 8 m

• The top of tree touches the ground makes an angle of 30°

$\underline{\underline{\sf{\maltese\:\:To\:Find}}}$

• Height  of the Tree

$\underline{\underline{\sf{\maltese\:\:Answer}}}$

• Height of Tree = 8√3 meters

$\underline{\underline{\sf{\maltese\:\:Calculations}}}$

• Let BD be the tree broken at point C such that the broken part CD takes the position CA and strikes the ground at A.

• It is given that AB = 8 m and ∠BAC = 30°

♣ Let BC= x meters and CD = CA = y meters

So in ΔABC, We Have :

• tan 30° = BC/AB

Since We considered BC = x meters

⇒ tan 30° = x/AB

⇒ sin 30°/cos 30° = x/AB

⇒ (1/2)/(√3/2) = x/AB

⇒ (1 × 2)/(√3 × 2) = x/AB

Cancelling 2 in numerator and denominator

⇒ 1/√3 = x/AB

It is already given AB = 8 m

⇒ 1/√3 = x/8

Cross Multiply

⇒ √3 × x = 8 × 1

⇒ √3x = 8

Dividing both sides by √3

⇒ √3x/√3 = 8/√3

⇒ x = 8/√3

Again in ΔABC, We Have :

• cos 30° = AB/AC

Already given AB = 8 meters

⇒ sin (90° - 30°) = 8/AC

⇒ sin (60°) = 8/AC

⇒ √3/2 = 8/AC

Since We considered AC or CA = y meters

⇒ √3/2 = 8/y

Cross Multiply

⇒ √3 × y = 2 × 8

⇒ √3y = 16

Dividing both sides by √3

⇒ √3y/√3 = 16/√3

⇒ y = 16/√3

______________________________________

Height of Tree = (x + y) meters

⇒ Height of Tree = (8/√3 + 16/√3) meters

⇒ Height of Tree = ((8 + 16)/√3) meters

⇒ Height of Tree = (24)/√3) meters

⇒ Height of Tree = ((2³ × 3) /√3) meters

⇒ Height of Tree = ((2³ × √3 × √3) /√3) meters

⇒ Height of Tree = ((2³ × √3)/1) meters

⇒ Height of Tree = 2³ × √3 meters

⇒ Height of Tree = 2³√3 meters

⇒ Height of Tree = 8√3 meters

Answer 2

Given:-

• Angle of elevation made by broken top of the tree = 30°
• Distance between the top of the tree and foot of the tree = 8m

Find:-

• Height of the tree.

Diagram:-

Let, AB be the original height of the tree. Suppose it got bent from the point C and let that the part CB takes the position CD when touching the ground at Point D.

Then,

$\setlength{\unitlength}{1cm}\begin{picture}(6,5)\linethickness{.4mm}\qbezier(5.5,4.9)(5.5,4.5)(5.5,1)\qbezier(1,1)(5.5,1)(5.5,1) \thicklines\multiput(1,1.2)(1.2,1){4}{\line(1,1){0.5}}\multiput(5.5,4.2)(0,1){4}{\line(0,1){0.5}} \put(5.4,7.8){\bf B} \put(5.6,1){\bf A}\put(5.6,4.8){\bf C} \put(0.5,0.8){\bf D}\qbezier(1.2,1.4)(2,1.6)(1.7,1)\put(1.8,1.4){ \bf 30^{\circ} } \put(5.18,1.02){\framebox(0.3,0.3)}\put(2.2,3){\sf y m}\put(5.6,6.4){\bf y m} \put(5.6,2.5){\bf x m}\put(3,0.7){\bf 9 m} \end{picture}$

Note: Kindly, See this diagram from web.

Solution:-

Let, AC = 'x' m

and CD = CB = 'y' m

Here, In right ∆DAC

➜ P/B = AC/AD = tan30°

➜ AC/AD = tan30°

where,

• AC = 'x' m
• AD = 9m
• tan30° = 1/√3

☯ Substituting these values ☯

➨ AC/AD = tan30°

➨ x/9 = 1/√3

• Cross-multiplication •

➨ √3x = 9

➨ x = 9/√3

• Rationalising The Denominator •

➨ x = 9/√3 × √(3)/√(3)

➨ x = 9√3/3

➨ x = 3√3 m

$\qquad$__________________

Now, again from right ∆DAC

➤ H/B = CD/AD = sec30°

➤ CD/AD = sec30°

where,

• CD = 'y' m
• AD = 9m
• sec30° = 2/√3

☯ Substituting these values ☯

➱ CD/AD = sec30°

➱ y/9 = 2/√3

• Cross-multiplication •

➱ √3y = 9×2

➱ √3y = 18

➱ y = 18/√3

• Rationalising The Denominator •

➱ y = 18/√3 × √(3)/√(3)

➱ y = 18√3/3

➱ y = 6√3 m

$\qquad$__________________

Total Height of the tree = AC + BC = x + y

where,

• x = 3√3m
• y = 6√3m

☯ Substituting these values ☯

➮ Total height of the tree = 3√3 + 6√3

➮ Total height of the tree = 9√3m

Hence, the total height of the tree will be 9√3m