This particular question is a combined case of physics and trigonometry ,where shell is being projected at an angle,means we have to apply the concept of projectile motion of an object.
the diagrammatic presentation of the problem is attached as an image.
Let O be the point of projection.
Let the shell be projected with an initial velocity of u m/sec at an angle tan−131=α(let) and it reached point A ,60 m short of the target point T.So in this case range of the projectile motion is OA.
Secondly when the shell is fired with an initial velocity of u m/sec at an angle of 45° ,it reached point B, 80 m beyond the target point T.
So in this case range of projectile motion is OB.
Range in projectile motion is given by
R=gu2sin2α where g is an acceleration due to gravity.
CASE I : In this case (tan−131=α⟹tanα=31)
OA=gu2sin2αNow sin2α=1+tan2α2tanα ................(1)tanα=31putting the value of tanα=31 in(1)we get sin2α=1+912∗31=53.OA=gu2∗53OA=5g3u2
similarly
CASE II: In this case
OB=gu2sin(2∗45)°OB=gu2sin90°=gu2 (∵ sin90°=1)
NowOA=53∗gu2OA=53∗OB (∵ OB=gu2)Now let OA=x meters.∴x=53∗(x+140)or,x=53x+420or,5x=3x+420or,5x−3x=420or,2x=420or,x=2420=210 meters.∴OA=210 meters.
Now from the above diagram we can see target is at a distance of OT meters from the point of projection.
∴OT=OA+ATor,OT=210+60=270 meters.
ANSWER: The target is 270 meters away from the point of projection.
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