Question

An aeroplane when flying at a height of 4000 m from the ground passes vertically above another aeroplane at an
instant when the angles of the elevation of the two planes from the same point on the ground are 60° and 45°
respectively. Find the distance between the aeroplanes at that instant.​

Answer 1

Answer:

$\large\bold\red{\frac{4( \sqrt{3} - 1) }{ \sqrt{3} } \: Km}$

Step-by-step explanation:

Given that,

An aeroplane is above another plane at an instant having angle of elevation 60° and 45° respectively.

Let ,

the planes are at pt. A and B above the point C , which is just below the aeroplanes.

and,

the angle of elevation is measured from pt. O.

Note:- Refer to attachment for the figure.

Now,

We have,

AC = 4000 m

Let,

AB = 'x' m

Therefore,

BC = (4000 - x) m

Now,

We have,

In ∆OBC,

$\tan(45) = \frac{4000 - x}{OC} \: \: \: \: \: \: \: \: \: ..........(i)$

Also,

In ∆OAC,

$\tan(60) = \frac{4000}{OC} \: \: \: \: \: \: \: \: .................(ii)$

Now,

dividing Equation $(i)\;and\;(ii)$,

We get,

$= > \frac{ \tan(45) }{ \tan(60) } = \frac{ \frac{4000 - x}{OC} }{ \frac{4000}{OC} } \\\\ = > \frac{ \tan(45) }{ \tan(60) } = \frac{4000 - x}{4000}$

But,

We know that,

$\tan(45) = 1 \: \: \: \: and \: \: \: \: \tan(60) = \sqrt{3}$

Therefore,

Putting the values,

We get,

$= > \frac{1}{ \sqrt{3} } = \frac{4000 - x}{4000} \\ \\ = > 4000 = \sqrt{3} (4000 - x) \\\\ = > 4000 = 4000 \sqrt{3} - \sqrt{3} x \\\\ = > 4000 \sqrt{3} - 4000 = \sqrt{3} x \\ \\ = > \sqrt{3} x = ( \sqrt{3} - 1)4000 \\ \\ = > x = \frac{4000( \sqrt{3} - 1)}{ \sqrt{3} } \: m \\\\ = > x = \frac{4000 ( \sqrt{3 } - 1)}{1000 \sqrt{3} } \: Km \\\\ = > x = \frac{4( \sqrt{3} - 1)}{ \sqrt{3} } \: Km$

Hence,

the distance between the aeroplanes at that instant is $\frac{4( \sqrt{3} - 1) }{ \sqrt{3} } \: Km$

$\bold\red{Note:-}$

• Here, in the Question it's not given to use the value of$\sqrt{3}$.

• That's why we can't take value of $\sqrt{3}$ until it's given.

• And if you think we can take values, then it should be kept kin mind what value we will use, 1.7 or 1.73 or 1.732 or 1.7321 or upto what decimal places ??

• So, U can't take a specific value of $\sqrt{3}$here.