let's make a diagram.
(I think your's is probably close)
draw a straight line and mark two points A and B and label AB = 1 km
Place the plane above the line AB and to the right of it, call it P
Extend AB, and from P draw a perpendicular down to the extended line AB and let Q be the point where it meets AB extended.
You should now have two right-angled triangles, APQ and BPQ, with the right angle at Q.
Label BQ as x, and PQ as y
now in the inside triangle,
tan 24.7º = y/x
y = xtan24.7
in the larger triangle,
tan 22.8 = y/(x+1)
y = (x+1)tan22.8
so (x+1)tan22.8 = xtan24.7
xtan22.8 + tan22.8 = xtan24.7
tan22.8 = xtan24.7 - xtan22.8
tan22.8 = x(tan24.7 - tan22.8)
x = tan22.8/(tan24.7-tan22.8)
now use your calculator to find x (I don't mine handy)
once you have x, sub it back into
y = xtan24.7 to find the height y
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