Question

Two sides of a triangle are 8 m and 9 m in length and the angle between them is increasing at a rate of 0.06 rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length isπ3rad.

Answer 1

Rate at which area is increasing is 1.08 m²/s.

Step-by-step explanation:

Area of triangle is with side a and b and angle C between them is given by

                             A = 0.5 ab SinC

Here we need to find how area changes with  a and b fixed and C is changing,

                       $\frac{dA}{dt}=\frac{d}{dt}\left (0.5 absinC\right )\\\\\frac{dA}{dt}=0.5ab\frac{d}{dt}\left (sinC\right )\\\\\frac{dA}{dt}=0.5abcosC\frac{dC}{dt}$

We have

                 a = 8 m

                 b = 9 m

                 $C=\frac{\pi}{3}rad\\\\\frac{dC}{dt}=0.06rad/s$

Substituting

                 $\frac{dA}{dt}=0.5\times 8\times 9\times cos\left ( \frac{\pi}{3}\right )\times 0.06\\\\\frac{dA}{dt}=1.08m^2/s$

Rate at which area is increasing is 1.08 m²/s.