# A plane flies from base camp to lake A, a distance of 280 km at a direction of 20.0° north of east. After dropping off supplies, the plane flies to lake B, which is 190 km and 30.0° west of...

A plane flies from base camp to lake A, a distance
of 280 km at a direction of 20.0° north of east. After
dropping off supplies, the plane flies to lake B, which
is 190 km and 30.0° west of north from lake A. Graphi-
cally determine the distance and direction from lake B
to the base camp.

This is confusing to me and I would really appreciate a detailed answer. I worked this problem out and I acquired 214.7 Km for distance and the direction is 62.68 degrees West of North. The answer key has the answer as 310 km at 57 degrees South of East. I don't know where I went wrong. Thank you.

flbyrne | Certified Educator

See the picture for the graphical solution. The distance of 310 km. is good but the direction is 57 degrees North of East.

You can verify using trigonometry.

Law of cosines:

`d^2=(280)^2+190^2-(2)(280)(190)cos80=310 `

Law of sines:

`sin (alpha)/190=sin80/310`

`sin(alpha)=(190sin80)/310=0.6`

`alpha=37`

The angle from East is: `20+alpha=57` degrees

Thus the distance from base camp to Lake B is 310 km and the direction is 57 degrees North of East.

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