Answer:
0.339 m/min
Explanation:
You want to know the rate of change of the third side in a triangle with sides 10 m and 14 m and an angle between them of 60°. The angle is increasing at 2°/min.
Law of Cosines
The law of cosines tells you the third side (c) satisfies the relation ...
c² = a² +b² -2ab·cos(C)
Filling in the given values, we have ...
c² = 10² +14² -2(10)(14)cos(C) = 296 -280cos(C)
Rate of change
Taking the square root and differentiating with respect to time, we have ...
c = √(296 -280cos(C))
c' = 280sin(C)·C'/(2√(296 -280cos(C)))
We want the value of this when C=60°, and C' = 2°/min = π/90 rad/min.
c' = 280(sin(60°))·(π/90)/(2√(296 -280cos(60°))) = (14π√3/9)/(2√156)
c' ≈ 0.339 . . . . m/min
The third side is increasing at a rate of about 0.339 meters per minute.