Please Help!! Finding the length of the missing side in the triangle. 10 points!?

You can use law of cosines. You have a known length opposite of the known angle, so that is your "c" and "C". Your unknown is then either "a" or "b", won't matter which. I'll chose "b". Also note the design is not to scale. With the bottom length longer than the other given length, its opposite angle has to be larger than the given angle. Leaving the third angle to be very small, resulting in your unknown length to be very small in comparison:

c² = a² + b² - 2ab cos(C)
29.5² = 30² + b² - 2(30)b cos(71)
870.25 = 900 + b² - 60b cos(71)
-29.75 = b² - 60b(0.325568)
0 = b² - 19.53408b + 29.75

quadratic equation:

b = [ -b ± √(b² - 4ac)] / (2a)
b = [ -(-19.53408) ± √((-19.53408)² - 4(1)(29.75))] / (2(1))
b = [ 19.53408 ± √(381.5802814464 - 119)] / 2
b = [ 19.53408 ± √(262.5802814464)] / 2
b = (19.53408 ± 16.20433) / 2

So we have two possible values for "b":

b = 3.32975 / 2 and b = 35.73841 / 2
b = 1.7 mm and b = 17.9 mm

So there are two possible solutions.

http://oi64.tinypic.com/bevrqa.jpg

Side 1: 30 opposite angle: 74.059°
Side 2: 29.5 opposite angle: 71°
Side 3: 17.869 opposite angle: 34.941°
Total Area: 253.43511741082
http://www.calculator.net/triangle-calculator.html?vc=&vx=30.00&vy=29.50&va=71&vz=&vb=&angleunits=d&x=67&y=8