Rate of Change Question8

In this page rate of change question8 we are going to see solution of some practice question of the worksheet.

Question 8:

Two sides of a triangle have length 12 m and 15 m. The angle between them is increasing at a rate of 2°/min. How fast is the length of third side increasing when the angle between the sides of fixed length is 60°?

Solution:

Let ABC be the given triangle

Length of the side AB = 12 m

length of the sides BC = 15 m

Let θ be the angle between the sides AB and BC at "t" minutes

dθ/dt = 2°(π/180)'

        = π/90

now we have to find db/dt when θ = 60°

b² = c² + a² - 2 a c cos θ

2 b (db/dθ) = 0 + 0 - 2 a c (-sin θ) (dθ/dt)

2 b (db/dθ) =  2 a c (sin θ) (dθ/dt)

b (db/dθ) =  a c (sin θ) (dθ/dt)  --- (1)

now apply a = 15 , c = 12 and θ = 60°

b² = (12)² + (15)² - 2 (15) (12) cos 60°

b² = 144 + 225 - 360 (1/2)

b² = 369 - 180

b² = 189

b = √189

now let us apply these values in the first equation

b (db/dθ) =  a c (sin θ) (dθ/dt)  --- (1)

√189 (db/dθ) =  (15) (12)(sin 60°) (π/90)

(db/dθ) =  (15) (12)(√3/2) (π/90)/√189

(db/dθ) = [(180) (√3) (π)]/[2 (√189) (90)]

(db/dθ) = π/√63 m/minutes

Therefore the rate of change of the third side is (π/√63) m/minutes.