Heights and Distances
On a horizonal plane there is a vertical tower with a flagpole on the top of the tower. At a point, 9 meters away from the foot of the tower, the angle of elevation of the top and bottom of the flagpole are
and 
respectively. Find the height of the tower and the flagpole mounted on it15. The height of an observation tower is
above sea level. A ship coming towards the tower is observed at an angle of depression of 
. Calculate the distance of the boat from the foot of the observation tower.
As per data given in the question, Let us assume that PQ be the observation tower with height$180m$above sea level and R be the position of the ship. Also assume that the distance of the boat from...
14. Find the angle of depression from the top of a 
high pillar of a milestone on the ground at a distance of 
from the foot of the pillar.
According to the question, Let us assume that PQ be the pillar of milestone and also angle of depression from top of the pillar to ground be Consider the ΔPQR, $\tan \theta =\frac{PQ}{RQ}$...
13. The top of a palm tree having been broken by the wind struck the ground at an angle of 
at a distance of 
from the foot of the tree. Find the original height of the palm tree.
As per information given in the question, Let us assume that PR was original tree. It was broken into two parts by wind. The broken part P’Q is making ${{60}^{\circ }}$ with ground. In ΔP’QR,...
12. Due to a heavy storm, a part of a banyan tree broke without separating from the main. The top of the tree touched the ground
from the base making an angle of 
with the ground. Calculate the height of the tree before it was broken.
According to the question, Let us assume that PR was original tree and due to storm it was broken into two parts, the broken part P’Q is making ${{45}^{\circ }}$with ground. In ΔP’QR, $\tan...
11. A ladder rests against a tree on one side of a street. The foot of the ladder makes an angle of 
with the ground. When the ladder is turned over to rest against another tree on the other side of the street it makes an angle of 
with the ground. If the length of the ladder is ![\[60m\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a844a9f8fafcbec9496fb007ccea03e1_l3.png "Rendered by QuickLaTeX.com")
, find the width of the street.
As per given information in the question, Let us assume two trees are denoted by PQ and RS and O be a point on the street PR, which is the distance between the two trees. Then, We Assume that OS and...
10. A 
high pole is kept vertically by a steel wire. The wire is inclined at an angle of with the horizontal ground. If the wire runs from the top of the pole to the point on the ground where its other end is fixed, find the length of the wire.
According to the given question, Let us assume that the wire which runs from the top of the pole to the point on the ground where its other end is fixed is PR and pole is represented by PQ. In ΔPQR,...
9. The top of a ladder reaches a point on the wall 
above the ground. If the foot of the ladder makes an angle of ![\[{{30}^{\circ }}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-34cd53219eaf0763feb7ca785cfc2855_l3.png "Rendered by QuickLaTeX.com")
with the ground, find the length of the ladder.
As per information given in the question, Let us assume that PR be the ladder and PQ is the wall and the point which is $5m$ above the ground is represented by P Then, In ΔPQR, $\sin {{30}^{\circ...
8. The topmost branch of a tree is tied with a string attached to a pole in the ground. The length of this string is and it makes an angle of with the ground. Find the distance of the pole to which the string is tied from the base of the tree.
According to the question, Let us assume that the topmost branch of the tree is Q and the pole on the ground to which the topmost branch is tied is R In ΔPQR, $\cos {{45}^{\circ }}=\frac{PR}{PQ}$...
7. A kite tied to a string makes an angle of ![\[{{60}^{\circ }}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d51617462bf9c796603d772f1894cc80_l3.png "Rendered by QuickLaTeX.com")
with the ground. Find the perpendicular height of the kite if the length of its string is 
.
According to the question, Let us assume that P is the kite in sky and the string is tied to point R on ground. Then, In ΔPQR, \[\sin {{60}^{\circ }}=\frac{PQ}{PR}\]...
6. An aeroplane takes off at angle of
with the ground. Find the height of the aeroplane above the ground when it has travelled ![\[386m\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-8713ebf248c7671d9090425c001c7035_l3.png "Rendered by QuickLaTeX.com")
without changing direction.
According to the given in the question, Let us assume that the plane takes off from point R on the ground and also assume that P be the final position of the plane to landoff. Then, In ΔPQR, $\sin...
5. Find the length of the shadow cast by a tree 
high when the sun’s altitude is
.
Let us assume that PQ is height of the tree and QR is the shadow of tree. Then, In ΔPQR, \[\tan {{30}^{\circ }}=\frac{PQ}{QR}\] \[\frac{1}{\sqrt{3}}=\frac{60}{QR}\] \[QR=60\sqrt{3}m\] Therefore,...
4. The length of the shadow of a pillar is 
times the height of the pillar. Find the angle of elevation of the sun.
Let us assume PQ is the pillar and QR is the shadow of pillar. Let us assume ‘h’ be the height of the pillar. Then, length of the shadow of pillar is $\frac{1}{\sqrt{3}}h$ QR =...
3. A vertical pole is 
high and the length of its shadow is ![\[90\sqrt{3}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3920778c2464170a402135c2ef0152b7_l3.png "Rendered by QuickLaTeX.com")
. What is the angle of elevation of the sun?
Let us assume PQ be the pole and QR be the shadow of the pole. Then, In ΔPQR, \[\tan \theta =\frac{PQ}{QR}\] \[\tan \theta =\frac{90}{90\sqrt{3}}\] \[\tan \theta =\frac{1}{\sqrt{3}}\] So as we...
2. The angle of elevation of the top of a vertical cliff from a point ![\[30m\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1fdc10a5faa9e1d42c2f2b00747bfc8e_l3.png "Rendered by QuickLaTeX.com")
away from the foot of the cliff is . Find the height of the cliff.
Let us assume that PQ is the cliff and angle of elevation from R to the cliff is${{60}^{\circ }}$. In ΔPQR, \[\tan \theta =\frac{PQ}{QR}\] \[\tan {{60}^{\circ }}=\frac{PQ}{30}\]...
1. The distance of the gate of a temple from its base is ![\[\sqrt{3}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-fafa997384c9af1c9948d8c2c13a4054_l3.png "Rendered by QuickLaTeX.com")
times it height. Find the angle of elevation of the top of the temple.
Let us assume that PQ be the temple and R be the position of gate of the temple. Let us assume “h” be the height of the temple, Then, PQ = h QR = Distance of the gate of temple from its base...