Geometry

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In what direction must a plane head if it can maintain 350 km/h in still air, wishes to fly 500km in a diirection 040degree and the wind is blowing at 56km/h from 100 degree? howlong would the journey take(to the nearest minutes)? how long would the return journey take, again to the nearest minutes, if the wind is still 56km/h from 100 degree?

I drew the following diagram.
OX is the x-axis, and OY is the y-axis

OB is the desired vector you want to go
so angle(YOB)=40º, it has a magnitude of 500.
Let OA be the vector the plane has to fly, AB is the vector representing the direction and velocity of the wind.
(point A is to the right of B and slightly downwards so it makes a 10º angle with the horizonatal)

Using a bit of basic geometry it should be easy to follow that angle(OBA)=120º

If we let the time taken be t hours, we can label the maginitude of AB as 56t and the magnitude of OA as 350t

Using the cosine law I got the equation
(350t)^2 = 500^2 + (56t)^2 - 2(500)(56t)cos 120º
or
119364t^2 - 28000t - 250000=0
check my arithmetic but by the quadratic formula I got t=1.56 or (a negative t, which we reject)

So the time taken is 1.56 hours
=1.56*60 minutes
=94 minutes to the nearest minute.

Using the Sine Law, Sin(BOA)/(56*1.56) = sin 120/(350*1.56)
sin(BOA)=.1385
angle(BOA)=7.96º

so the direction should be 40º + 7.96º
or appr 48º

Setting up a similar geometrical figure you should be able to do the second part yourself.

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