Answer

Verified

387.6k+ views

**Hint:**

Here, we will solve this question of heights and distances by using the similarity of triangles. We will prove that the given two right angled triangles are similar to each other. Hence, their corresponding sides will be proportional. Equating the corresponding sides of similar triangles, and solving it further, will we get the required height of the tree.

**Complete step by step solution:**

Let \[BC\] be the height of Ravi.

According to the question, height of Ravi \[ = BC = 1.82{\rm{m}}\]

Now, let the height of the tree in Ravi’s backyard be \[DE = h\] meters

Now, it is given that from the tree’s base he walked \[12.20{\rm{m}}\].

Hence, distance covered by him, \[BD = 12.20{\rm{m}}\]

Also, it is gven that he is now \[6.10{\rm{m}}\] from the end of the shadow.

Therefore, from the figure, the distance \[AB = 6.10{\rm{m}}\]

Now, as we can see,

Ravi’s shadow \[ = AB = 6.10{\rm{m}}\]

And, Tree’s shadow \[ = AD = BD + AB\]

Substituting \[AB = 6.10{\rm{m}}\] and \[BD = 12.20{\rm{m}}\] in the above equation, we get

\[ \Rightarrow AD = 12.20 + 6.10\]

Adding the terms, we get

\[ \Rightarrow AD = 18.30{\rm{m}}\]

Now, in $\vartriangle ABC$ and $\vartriangle ADE$,

\[\angle ABC = \angle ADE = 90^\circ \] (As the height of Ravi and the tree will always be perpendicular towards the ground)

\[\angle A = \angle A\] (common angle)

Therefore, by Angle-Angle or AA Similarity

$\vartriangle ABC \simeq \vartriangle ADE$

Hence, if two triangles are similar, then, the corresponding parts of similar triangles are proportional. So,

\[\dfrac{{AB}}{{AD}} = \dfrac{{BC}}{{DE}}\]

Substituting \[DE = h\], \[AB = 6.10{\rm{m}}\], \[BC = 1.82{\rm{m}}\] and \[AD = 18.30{\rm{m}}\] in theabove equation, we get

\[ \Rightarrow \dfrac{{6.10}}{{18.30}} = \dfrac{{1.82}}{h}\]

Now, by cross multiplying, we get,

\[ \Rightarrow 6.10h = 1.82 \times 18.30\]

On converting decimal to fraction, we get

\[ \Rightarrow h = \dfrac{{182 \times 1830 \times 100}}{{100 \times 100 \times 610}}\]

Solving further, we get,

\[ \Rightarrow h = \dfrac{{546}}{{100}} = 5.46{\rm{m}}\]

Therefore, the required height of the tree is \[5.46{\rm{m}}\].

**Hence, the tree is \[5.46{\rm{m}}\] tall.**

**Note:**

An alternate way to solve this question is:

We will use trigonometric identities in a right angled triangle.

As we know, in a right angled triangle \[\tan \theta = \dfrac{P}{B}\], where \[P\] is the perpendicular side and \[B\] is the base.

Therefore, in $\vartriangle ABC$

\[\tan \theta = \dfrac{{BC}}{{AB}}\]

Substituting \[AB = 6.10{\rm{m}}\]and \[BC = 1.82{\rm{m}}\] in the above equation, we get

\[\tan \theta = \dfrac{{1.82}}{{6.10}}\]

Simplifying further, we get

\[\tan \theta = \dfrac{{91}}{{305}}\]……………………………. \[\left( 1 \right)\]

But, if we consider the larger right angled triangle, i.e. $\vartriangle ADE$,

Then, \[\tan \theta = \dfrac{{DE}}{{AD}} = \dfrac{h}{{18.30}}\]………………………….. \[\left( 2 \right)\]

Hence, equating \[\left( 1 \right)\] and \[\left( 2 \right)\], we get,

\[\dfrac{h}{{18.30}} = \dfrac{{91}}{{305}}\]

On cross multiplication, we get

\[ \Rightarrow h = \dfrac{{91 \times 1830}}{{305 \times 100}}\]

Simplifying the expression, we get

\[ \Rightarrow h = \dfrac{{546}}{{100}} = 5.46{\rm{m}}\]

Therefore, the required height of the tree is \[5.46{\rm{m}}\]

Hence, the tree is \[5.46{\rm{m}}\] tall.

Recently Updated Pages

Which are the Top 10 Largest Countries of the World?

Differentiate between Shortterm and Longterm adapt class 1 biology CBSE

How do you find slope point slope slope intercept standard class 12 maths CBSE

How do you find B1 We know that B2B+2I3 class 12 maths CBSE

How do you integrate int dfracxsqrt x2 + 9 dx class 12 maths CBSE

How do you integrate int left dfracx2 1x + 1 right class 12 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Name 10 Living and Non living things class 9 biology CBSE

The Buddhist universities of Nalanda and Vikramshila class 7 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE