Find the rate?

Question

Two sides of a triangle are 6 m and 12 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 is π/3 rad.

Answer 1

Let θ = the measure (in radians) of the angle at time t seconds and let A = area of triangle at time t seconds.

 

The area of a triangle is ½absinθ, where a and b are two of the sides of the triangle and θ is the angle between those two sides. 

 

Given:  dθ/dt = 0.06

 

Find:  dA/dt when θ = π/3

 

Solution:  A = ½(6)(12)sinθ = 36sinθ

 

              So, dA/dt = 36cosθ(dθ/dt)

 

                            = 36cos(π/3)(0.06)

 

                            = 36(½)(0.06)

 

                            = 1.08 m²/sec