two sides of a triangle have lengths 10 m and 12 m....

Question

two sides of a triangle have lengths 10 m and 12 m. the angle between them is increasing at a rate of .05 radians /min. how fast is the length of the third side increasing when the angle between the sides of fixed length is pi/3 radians

Answer 1

Use law of Cosines formula

c² = a² + b² - 2ab*Cos C

a=10 meters

b=12 meters

At pi/3

c² = 10² + 12² - 2(10)(12)*Cos (pi/3) = 100 + 144 - 240(.5) = 244 - 120 = 124

c = 2*sqrt(31)

Take derivative of the law of cosines formula with respect to time

2c*dc/dt = 2ab*Sin C*dC/dt

dc/dt = (ab/c)(Sin C)(dC/dt)

dC/dt = .05 radians/minute

dc/dt = (10 meters*12 meters)/(2sqrt(31) meters) *(Sin pi/3)*(0.05 radians/minute) = 0.467 meters/minute