A problem using the sine laws?

Question

A tree grows vertically on the side of a hill that slopes upward from the horizontal by 9 degrees. When the angle of elevation of the sun relative to the top of the tree measures 22 degrees, the shadow of the tree falls 44 feet down the hill. How tall is the tree?

Answer 1

Draw a diagram for this one.

Start with a right triangle whose right angle is to the lower left.  Mark the right angle "B", the upper left angle "A", and the lower right angle "C".  Label angle C as being 9 degrees.  AC represents the hill.  

Now pick the point in the middle of AC, and draw a veritcal line straight up from it.  This is the tree.  Mark the bottom of the tree (the midpoint of AC) as "D", and mark the top of the tree as "E".  The shadow is DC.

Draw line EC and keep extending it to the upper left to some arbitrary point "F".  This represents the sunlight coming down from the upper left (F) and down on to the tip of the tree (E), to the tip of the shadow (C).  Finally, draw a horizontal line starting at E and going to the left to some end point "G".  Angle GEF is the angle of elevation.  Angle DEG is a right angle.

Since GEF = 22, DEG=90, and CEF is a straight line.  That means DEC is 78 degrees.  Label it.  Angle BCE should also be 22 degrees, so angle ACE is 13.  Label it.

So you have triangle DEC where DE is the tree, and CD is the shadow.  By the Law of Sines,

sin(13) / DE = sin(78)/44

Solve this for DE.

Angle DEF is goin to be a 90 degree angle plus the angle of elevation.

Answer 2

Horizontal distance from tree to end of shadow = d

d = 44 * cos9°

Vertical distance from top of tree to end of shadow = z

z = d * tg31°

Height of the tree = z - (44 * sin9°) = 19.23 ft

Answer 3

using sine law

x/sin(22-9) = 44/sin(180-90-22)

x = 44*sin(13)/sin(68)

x = 10.68 ft