To a cyclist riding west at 20 kg/hr, the rain appears to meet him at an angle of 45° with the vertical. when he rides at 12 km/hr, the rain meets him at an angle of 19°48’ with the vertical. what is the actual direction of the rain?

Get the angle between the vertical and the line of sight of the cyclist parallel to the rain at 19°48’ 
180 - 90 - 19°48’  = 70.2°
Prepare two equations to solve for the velocity of the rain
DC2 sin 70.2 = (12 - x) tan 70.2
DC1  tan 45 = (20 - x) tan 45

Solving the equation results to
x = 3.8
and
RD = 14.20

The actual angle of the rain is,
90 - tan-1(14.20/3.8) = 15°

Neptune is the eighth planet from the sun. it was the first planet to get its existence predicted by  mathematical calculations  before it was actually seen through a telescope on sept. 23, 1846. irregularities in the orbit of  uranus  led french astronomer alexis bouvard to suggest that the gravitational pull from another celestial body might be responsible. german astronomer johann galle then relied on subsequent calculations to   spot neptune via telescope. previously, astronomer galileo galilei sketched the planet, but he mistook it for a star due to its slow motion. in accordance with all the other planets seen in the sky, this new world was given a name from greek and roman mythology — neptune, the roman god of the sea.

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