Rate of Change Question7

In this page rate of change question7 we are going to see solution of some practice question of the worksheet.

Question 7:

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.

Solution:

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