Precalculus Problems Help?

Question

Sketch the plot of land described below and find its area to the nearest square unit:
From the intersection of 2nd Street and 3rd street, proceed N32 W for 350m along 2nd street, then S56 W for 300m to the football field, then S22 E until 3rd street is reached, and finally N68 E along 3rd street back to the starting point.

Sketch the diagram:
A plane flies at a bearing of 75 degrees from departure point A to point B at 350 mph for 90 minutes. At point B, the plane adjusts bearing to 125 degrees and flies at 325 mph for 1 hour to point C. Find the bearing the plane must follow, and the distance to travel from point C directly back to point A.

Answer 1

For the first part you have a quadrilateral where you know all four angles and two of the sides.

Let the points along the quadrilateral be A, B, C, and D. with A being the starting point

Now look at:
http://www.btinternet.com/~se16/hgb/triangle.htm

Since you know sides AB and BC, as well as the angle at C, you can find the area of triangle ABC by using formula #2 for side-angle-side.

You can also compute the length of line AC in several ways.

(One is to put B at the origin, use distance and bearing (r and theta) to compute the x and y coordinates for A and C and then compute the distance between them)

With the distance, and sin(B), you can use the law of sines to compute the sine of angles BAC and BCA.

http://en.wikipedia.org/wiki/Law_of_sines

From the sines you can compute the angles themselves, and then angles DAC and DCA.

Add the areas of the two triangles and you're done.

Here are some other quadrilateral formulas, but they don't apply directly in this case:
http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node23.html

With the two angles and the common side, you can use formula #3 to compute the area of triangle DAC.

As for the second part, it is basically the same approach as you used to compute the length of AC in part 1: compute the x and y coordinates of A and C based on their bearing and distance from B. Then use the slope of the vector CA to get the angle.