Question

The angle of elevation of an airplane from a point A on the ground is 60° after a flight of 30 seconds, the angle of elevation changes to 30°. If the plane is flying at a constant height of 3600 root3m, find the speed, in km/hour, of the plane.

Answer 1

Given :

• The angle of elevation of a plane from a point A on ground is 60°
• After 30 seconds of flight, the angle of elevation becomes 30°
• Height at which the plane is flying = 3600√3 m.

To Find :

• Speed of the plane in km/hr.

Solution :

Let's first consider Δ ABC.

Angle of elevation = $\theta$ = 60°

BC = 3600√3 m.

Using,

tan $\theta$ = $\sf{\dfrac{Opposite\:side\:angle\:\theta}{Adjacent\:side\:to\:angle\:\theta}}$

$\theta$ = 60°

Opposite side = BC = 3600√3 m.

Adjacent side = AB

Block in the value,

$\longrightarrow$ $\sf{60\:^\circ\:=\:\dfrac{BC}{AB}}$

$\longrightarrow$ $\sf{\sqrt{3}=}$$\sf{\dfrac{3600\:\sqrt{3}}{AB}}$

$\longrightarrow$ $\sf{AB\:\sqrt{3}\:=\:3600\:\sqrt{3}}$

$\longrightarrow$ $\sf{AB\:=\:\dfrac{3600\sqrt{3}}{\sqrt{{3}}}}$

$\longrightarrow$ $\sf{AB\:=\:3600\:\:\:(1)}$

Now, consider the Δ EDA.

Angle of elevation = 30° = $\theta$

ED = 3600√3 m.

Again we will use $\sf{Tan\:\theta}$

Opposite side = ED = 3600√3 m

Adjacent side = DA

Block in the values,

$\longrightarrow$ $\sf{30\:^\circ\:=\:\dfrac{ED}{DA}}$

$\longrightarrow$ $\sf{\dfrac{1}{\sqrt{3}}=}$ $\sf{\dfrac{3600\sqrt{3}}{DA}}$

$\longrightarrow$ $\sf{DA=3600\:\sqrt{3}\:\times\:\sqrt{3}}$

$\longrightarrow$ $\sf{DA=\:3600\:\times\:3}$

$\longrightarrow$$\sf{DA\:=\:10800}$

Now, DA is the distance covered by the plane.

$\longrightarrow$ $\sf{DA=DB+AB}$

$\longrightarrow$$\sf{10800=DB+3600}$

$\longrightarrow$ $\sf{10800-3600=DB}$

$\longrightarrow$ $\sf{7200=DB}$

•°• The plane travelled a distance of 7200 m.

We have the time taken by the plane to cover this distance.

Time = 30 second.

Now, we know the formula of speed.

Formula :

$\sf{Speed\:=\dfrac{Distance}{Time}}$

Block in the values,

$\longrightarrow$ $\sf{Speed\:=\:\dfrac{7200}{30}}$

$\longrightarrow$ $\sf{Speed\:=\:240}$

The speed of plane per second is 240 m/s.

We have to change the units.

In 1 hr we have 3600 seconds.

Multiply the speed of plane per second by the total number of second in one hour.

$\longrightarrow$ $\sf{3600\:\times\:240}$

$\longrightarrow$ $\sf{864000}$

•°• Plane covers 864000 m in 3600 seconds i.e in 1 hr.

1 km = 1000 m.

•°• $\sf{\dfrac{864000}{1000}}$

$\longrightarrow$ $\sf{864\:km}$

•°• $\sf{Speed\:=\:\dfrac{Distance}{Time}}$

$\longrightarrow$$\sf{Speed=\dfrac{864}{1}}$

$\longrightarrow$ $\sf{Speed\:=\:864\:km/hr}$

$\large{\boxed{\sf{\red{Speed\:of\:plane\:in\:km\:/hr\:=\:864\:km/hr}}}}$