Question

Two sides of a triangle have lengths 9 m and 13 m. The angle between them is increasing at a rate of 0.02radians /min. How fast is the length of the third side increasing when the angle between the sides of fixed length is π/3 radians.

Answer 1

Answer:0.175 m/s

Step-by-step explanation:

Given

Two sides of length 9 and 13 m

angle between them is increasing at an angle of 0.02 rad/min

rate of increasing of third at $\theta =\frac{\pi }{3}$

let a=9

b=13

c=unknown side

Now using cos rule of triangle

$cos\theta =\frac{a^2+b^2-c^2}{2ab}$

$cos\theta =\frac{a^2+b^2}{2ab}-\frac{c^2}{2ab}$

Differentiating both sides we get

$-\sin \theta \times \frac{\mathrm{d} \theta }{\mathrm{d} t}=0-2\frac{c}{2ab}\times \frac{\mathrm{d} c}{\mathrm{d} t}$

at $\theta =\frac{\pi }{3}$

c=11.53 m

substituting values we get

$\frac{\sqrt{3}}{2}\times 0.02=\frac{11.53}{9\times 13}\times \frac{\mathrm{d} c}{\mathrm{d} t}$

$\frac{\mathrm{d} c}{\mathrm{d} t}=0.175 m/s$