The area of a circle is 78.5 square centimeters, and a subtending arc on the circle has an arc length of 6. the estimated value of is 3.14.
the measure of the angle subtended by the arc is __ degrees

I'm sure there are easier ways, but the way I'd do it is find the radius of the circle:

A=πr²
78.5=3.14r²
25=r²
5=r

Then find the circumference: 
C=2πr
C=2π5
C=31.4

then find the ratio of an arc with lenth of 6 to the whole circumference:
6/31.4=0.19108280254777070063694267515924
So an arc of length 6 is 19.108...% of the whole circle, so it would "subtend" the same percentage of degrees out of the circle's 360:

360*0.19108...=68.789808917197452229299363057325 degrees.

I'd put 68.79 degrees as my answer.