A jet plane is at a vertical height of h- The angles of depression of two tanks on the horizontal ground are found to have measures - and - - - Prove that the distance between the tanks is h tan- -tan- tan- -tan- assuming both the tanks are on the same side of the jet plane-

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Calculating Heights and Distances

A jet plane i...

Question

A jet plane is at a vertical height of $h$ . The angles of depression of two tanks on the horizontal ground are found to have measures $α$ and $β, α > β$ . Prove that the distance between the tanks is $\frac{h (tan α - tan β)}{tan α - tan β}$ assuming both the tanks are on the same side of the jet plane.

Open in App

Solution

Given that, a jet plane is at a vertical height of $h$ . And the angles of depression of two tanks on the horizontal ground are found to have measures $α$ and $β$ such that $α > β$ .
Based on the given information, we can draw the figure given above.
Let $A$ be the position of the jet plane and $C$ and $D$ be the two tanks on the same side of the jet plane
Hence, $A B = h$
$B C =$ distance between base and $1^{s t}$ tank $= y$
$C D =$ distance between two tanks $= x$

Now, in $△ A B C$
$tan α = \frac{A B}{B C}$
$∴ tan α = \frac{h}{y}$
$∴ y = \frac{h}{tan α} . . . (1)$
In $△ A B D$
$tan β = \frac{A B}{B D}$
$∴ tan β = \frac{h}{x + y}$
$∴ x + y = \frac{h}{tan β} . . . . (2)$
From (1) and (2)
$x + \frac{h}{tan α} = \frac{h}{tan β}$
$\Rightarrow x = \frac{h}{tan β} - \frac{h}{tan α}$

$\Rightarrow x = \frac{h (tan α - tan β)}{tan α tan β}$

Hence, the distance between the two tanks is $\frac{h (tan α - tan β)}{tan α tan β} [H e n c e p r o v e d]$

Suggest Corrections

Similar questions

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag-staff of height h. At a point on the plane, the angles of elevation of the bottom and the top of the flag-staff are $α$ and $β$ respectively. Find the height of the tower.

Q. Question 8
A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height "h" units. At a point on the plane, the angles of elevation of the bottom and the top of the flagstaff are

α

and

β

respectively. Prove that the height of the tower is

(\frac{h t a n α}{t a n β - t a n α})

Q. In the angle of elevation of a cloud from a point

h

metres above a lake be

β

, and the angle of depression of its reflection in the lake be

α

, prove that the height of the cloud is

h (\frac{tan α + tan β}{tan α - tan β})

From an aeroplane vertically above a straight horizontal road, the angles of depression of two consecutive mile stones on opposite sides of the aeroplane are observed to be $α a n d β$ . Show that the height in miles of aeroplane above the road is given by $\frac{t a n α t a n β}{t a n β + t a n α}$

Q. From the top of the lighthouse, the angles of depression of two stations on the opposite sides of it at a distance

d

apart are

α

and

β .

The height of the lighthouse is

Join BYJU'S Learning Program

A jet plane is at a vertical height of h. The angles of depression of two tanks on the horizontal ground are found to have measures α and β,α>β. Prove that the distance between the tanks is h(tanα−tanβ)tanα−tanβ assuming both the tanks are on the same side of the jet plane.