A bridge across a valley is 800 meters long. There is a temple in the valley directly below the bridge. The angle of depression of the top of the temple from the two ends of the bridge measures ${30^ \circ }$ and ${60^ \circ }$. Find the height of the bridge above the top of the temple.
Last updated date: 28th Mar 2023
•
Total views: 306.3k
•
Views today: 2.83k
Answer
306.3k+ views
Hint: Draw a diagram to clearly analyse the situation. Two right angled triangles will be formed for the two end points of the bridge. Calculate the height of the bridge above the top of the temple using one of the triangles and then put this value for another triangle.
Complete step-by-step answer:
Consider the above figure. A and B are two end points of the bridge.
The length of the bridge is given in the question as 800 meters. So, we have:
$ \Rightarrow AB = 800m .....(i)$
Next, C is the position of the temple. Angles of depression on the top of the temple from the end points of the temple are given as ${30^ \circ }$ and ${60^ \circ }$. So, we get:
$ \Rightarrow \angle CBA = {30^ \circ }{\text{ and }}\angle CAB = {60^ \circ }$
We have to calculate the height of the bridge above the top of the temple. From the figure, this height is CD. So, we have to calculate the length of CD. Let its value is h.
$ \Rightarrow CD = h .....(ii)$
Now, let $AD = x .....(iii)$
As we can see from the figure, $AD + DB = AB$
And from equation $(i)$, $AB = 800m$. So we have:
$
\Rightarrow AD + DB = AB, \\
\Rightarrow x + DB = 800, \\
\Rightarrow DB = 800 - x .....(iv) \\
$
Next, consider $\Delta ADC$,
$ \Rightarrow \tan {60^ \circ } = \dfrac{{CD}}{{AD}},$
We know that $\tan {60^ \circ } = \sqrt 3 $ and from equation $(ii)$ and $(iii)$, $CD = h$ and $AD = x$. Putting these values in above equation:
$
\Rightarrow \sqrt 3 = \dfrac{h}{x}, \\
\Rightarrow x = \dfrac{h}{{\sqrt 3 }} .....(v) \\
$
Now consider $\Delta BDC$,
$ \Rightarrow \tan {30^ \circ } = \dfrac{{CD}}{{BD}}$
We know that $\tan {30^ \circ } = \dfrac{1}{{\sqrt 3 }}$ and from equation $(ii)$ and $(iii)$, $CD = h$ and $DB = 800 - x$. Putting these values in above equation:
$
\Rightarrow \dfrac{1}{{\sqrt 3 }} = \dfrac{h}{{800 - x}}, \\
\Rightarrow 800 - x = h\sqrt 3 \\
$
Putting $x = \dfrac{h}{{\sqrt 3 }}$ from equation $(v)$, we’ll get:
$
\Rightarrow 800 - \dfrac{h}{{\sqrt 3 }} = h\sqrt 3 , \\
\Rightarrow h\left( {\sqrt 3 + \dfrac{1}{{\sqrt 3 }}} \right) = 800, \\
\Rightarrow h\left( {\dfrac{{3 + 1}}{{\sqrt 3 }}} \right) = 800, \\
\Rightarrow h = \dfrac{{800\sqrt 3 }}{4}, \\
\Rightarrow h = 200\sqrt 3 \\
$
Thus, the height of the bridge above the top of the temple is $200\sqrt 3 $ meters.
Note: Whenever angle of elevation or angle of depression is given, we use to analyse the scenario using a right angled triangle with the help of suitable trigonometric ratios depending upon what is given and what is asked in the question.
Complete step-by-step answer:

Consider the above figure. A and B are two end points of the bridge.
The length of the bridge is given in the question as 800 meters. So, we have:
$ \Rightarrow AB = 800m .....(i)$
Next, C is the position of the temple. Angles of depression on the top of the temple from the end points of the temple are given as ${30^ \circ }$ and ${60^ \circ }$. So, we get:
$ \Rightarrow \angle CBA = {30^ \circ }{\text{ and }}\angle CAB = {60^ \circ }$
We have to calculate the height of the bridge above the top of the temple. From the figure, this height is CD. So, we have to calculate the length of CD. Let its value is h.
$ \Rightarrow CD = h .....(ii)$
Now, let $AD = x .....(iii)$
As we can see from the figure, $AD + DB = AB$
And from equation $(i)$, $AB = 800m$. So we have:
$
\Rightarrow AD + DB = AB, \\
\Rightarrow x + DB = 800, \\
\Rightarrow DB = 800 - x .....(iv) \\
$
Next, consider $\Delta ADC$,
$ \Rightarrow \tan {60^ \circ } = \dfrac{{CD}}{{AD}},$
We know that $\tan {60^ \circ } = \sqrt 3 $ and from equation $(ii)$ and $(iii)$, $CD = h$ and $AD = x$. Putting these values in above equation:
$
\Rightarrow \sqrt 3 = \dfrac{h}{x}, \\
\Rightarrow x = \dfrac{h}{{\sqrt 3 }} .....(v) \\
$
Now consider $\Delta BDC$,
$ \Rightarrow \tan {30^ \circ } = \dfrac{{CD}}{{BD}}$
We know that $\tan {30^ \circ } = \dfrac{1}{{\sqrt 3 }}$ and from equation $(ii)$ and $(iii)$, $CD = h$ and $DB = 800 - x$. Putting these values in above equation:
$
\Rightarrow \dfrac{1}{{\sqrt 3 }} = \dfrac{h}{{800 - x}}, \\
\Rightarrow 800 - x = h\sqrt 3 \\
$
Putting $x = \dfrac{h}{{\sqrt 3 }}$ from equation $(v)$, we’ll get:
$
\Rightarrow 800 - \dfrac{h}{{\sqrt 3 }} = h\sqrt 3 , \\
\Rightarrow h\left( {\sqrt 3 + \dfrac{1}{{\sqrt 3 }}} \right) = 800, \\
\Rightarrow h\left( {\dfrac{{3 + 1}}{{\sqrt 3 }}} \right) = 800, \\
\Rightarrow h = \dfrac{{800\sqrt 3 }}{4}, \\
\Rightarrow h = 200\sqrt 3 \\
$
Thus, the height of the bridge above the top of the temple is $200\sqrt 3 $ meters.
Note: Whenever angle of elevation or angle of depression is given, we use to analyse the scenario using a right angled triangle with the help of suitable trigonometric ratios depending upon what is given and what is asked in the question.
Recently Updated Pages
If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

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

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE
