In this page rate of change question7 we are going to see solution of some practice question of the worksheet.
Two sides of a triangle are 4 m and 5 m in length and the angle between them is increasing at the rate of 0.06 rad/sec. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is π/3.
Let "b" and "h" be the base and height of the triangle ABC
Let θ radian be the angle between the sides AB and AC when b = 5 cm, c = 4 cm
dθ/dt = 0.06 radian/sec. Let "A" be the area of the triangle at time "t". From this we have to find the value of dA/dt when θ = π/3
Area of triangle ABC (A) = (1/2) b c sin θ
differentiating the whole equation with respect to "t"
dA/dt = (1/2) b c cos θ (dθ/dt)
at θ = π/3
= (1/2) (5) (4) (sin π/3) (0.06)
= (1/2) (20) (1/2) (0.06)
= (20/4) (0.06)
= 0.3 m²/sec
Therefore the area is increasing at the rate of 0.03 m²/sec