From the top of a building \( AB, 60 \mathrm{~m} \) high, the angles of depression of the top and bottom of a vertical lamp post \( CD \) are observed to be \( 30^{\circ} \) and \( 60^{\circ} \) respectively. Find the difference between the heights of the building and the lamp post.
Given:
From the top of a building \( A B, 60 \mathrm{~m} \) high, the angles of depression of the top and bottom of a vertical lamp post \( C D \) are observed to be \( 30^{\circ} \) and \( 60^{\circ} \) respectively.
To do:
We have to find the difference between the heights of the building and the lamp post.
Solution:
From the figure,
$\mathrm{AB}=60 \mathrm{~m}, \angle \mathrm{BDE}=30^{\circ}, \angle \mathrm{BCA}=60^{\circ}$
Let the horizontal distance between \( A B \) and \( C D \) be $\mathrm{AC}=x \mathrm{~m}$ and the height of the lamp post be $\mathrm{CD}=h \mathrm{~m}$.
This implies,
$\mathrm{AE}=\mathrm{CD}=h \mathrm{~m}$
$\mathrm{DE}=\mathrm{CA}=x \mathrm{~m}$
$\mathrm{BE}=60-h \mathrm{~m}$
We know that,
$\tan \theta=\frac{\text { Opposite }}{\text { Adjacent }}$
$=\frac{\text { BE }}{DE}$
$\Rightarrow \tan 30^{\circ}=\frac{60-h}{x}$
$\Rightarrow \frac{1}{\sqrt3}=\frac{60-h}{x}$
$\Rightarrow x=(60-h)\sqrt3 \mathrm{~m}$............(i)
Similarly,
$\tan \theta=\frac{\text { Opposite }}{\text { Adjacent }}$
$=\frac{\text { BA }}{CA}$
$\Rightarrow \tan 60^{\circ}=\frac{60}{x}$
$\Rightarrow \sqrt3=\frac{60}{(60-h)\sqrt3}$ [From (i)]
$\Rightarrow [(60-h)\sqrt3]\sqrt3=60 \mathrm{~m}$
$\Rightarrow (60-h)3=60 \mathrm{~m}$
$\Rightarrow 60-h=20 \mathrm{~m}$
$\Rightarrow h=60-20 \mathrm{~m}$
$\Rightarrow h=40 \mathrm{~m}$
$\Rightarrow x=(60-40)(1.73)=20(1.73)=34.64 \mathrm{~m}$
$\Rightarrow \mathrm{AB}-\mathrm{CD}=60-40=20 \mathrm{~m}$
Therefore, the difference between the heights of the building and the lamp post is $20 \mathrm{~m}$.
Related Articles
- From the top of a building \( A B, 60 \mathrm{~m} \) high, the angles of depression of the top and bottom of a vertical lamp post \( C D \) are observed to be \( 30^{\circ} \) and \( 60^{\circ} \) respectively. Find the horizontal distance between \( A B \) and \( C D \).
- The angles of elevation and depression of the top and the bottom of a tower from the top of a building, $60\ m$ high, are $30^{o}$ and $60^{o}$ respectively. Find the difference between the heights of the building and the tower and the distance between them.
- The angles of depression of the top and bottom of \( 8 \mathrm{~m} \) tall building from the top of a multistoried building are \( 30^{\circ} \) and \( 45^{\circ} \) respectively. Find the height of the multistoried building and the distance between the two buildings.
- From the top of a \( 50 \mathrm{~m} \) high tower, the angles of depression of the top and bottom of a pole are observed to be \( 45^{\circ} \) and \( 60^{\circ} \) respectively. Find the height of the pole..
- From a point on the ground the angles of elevation of the bottom and top of a transmission tower fixed at the top of \( 20 \mathrm{~m} \) high building are \( 45^{\circ} \) and \( 60^{\circ} \) respectively. Find the height of the transimission tower.
- From the top of a building \( 15 \mathrm{~m} \) high the angle of elevation of the top of a tower is found to be \( 30^{\circ} \). From the bottom of the same building, the angle of elevation of the top of the tower is found to be \( 60^{\circ} \). Find the height of the tower and the distance between the tower and building.
- The angle of elevation of the top of the building from the foot of the tower is \( 30^{\circ} \) and the angle of the top of the tower from the foot of the building is \( 60^{\circ} \). If the tower is \( 50 \mathrm{~m} \) high, find the height of the building.
- From the top of a \( 7 \mathrm{~m} \) high building, the angle of elevation of the top of a cable tower is \( 60^{\circ} \) and the angle of depression of its foot is \( 45^{\circ} . \) Determine the height of the tower.
- The angles of elevation of the top of a rock from the top and foot of a \( 100 \mathrm{~m} \) high tower are respectively \( 30^{\circ} \) and \( 45^{\circ} \). Find the height of the rock.
- Two men on either side of the cliff \( 80 \mathrm{~m} \) high observes the angles of elevation of the top of the cliff to be \( 30^{\circ} \) and \( 60^{\circ} \) respectively. Find the distance between the two men.
- From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a $20\ m$ high building are $45^o$ and $60^o$ respectively. Find the height of the tower.
- A \( 1.5 \mathrm{~m} \) tall boy is standing at some distance from a \( 30 \mathrm{~m} \) tall building. The angle of elevation from his eyes to the top of the building increases from \( 30^{\circ} \) to \( 60^{\circ} \) as he walks towards the building. Find the distance he walked towards the building.
- On the same side of a tower, two objects are located. When observed from the top of the tower, their angles of depression are \( 45^{\circ} \) and \( 60^{\circ} . \) If the height of the tower is \( 150 \mathrm{~m} \), find the distance between the objects.
- There are two temples, one on each bank of a river, just opposite to each other. One temple is $50\ m$ high. From the top of this temple, the angles of depression of the top and the foot of the other temple are \( 30^{\circ} \) and \( 60^{\circ} \) respectively. Find the width of the river and the height of the other temple.
- From the top of a 7 m high building, the angle of the elevation of the top of a tower is $60^{o}$ and the angle of the depression of the foot of the tower is $30^{o}$. Find the height of the tower.
Kickstart Your Career
Get certified by completing the course
Get Started