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 [tex]\theta =\frac{\pi }{3}[/tex]
let a=9
b=13
c=unknown side
Now using cos rule of triangle
[tex]cos\theta =\frac{a^2+b^2-c^2}{2ab}[/tex]
[tex]cos\theta =\frac{a^2+b^2}{2ab}-\frac{c^2}{2ab}[/tex]
Differentiating both sides we get
[tex]-\sin \theta \times \frac{\mathrm{d} \theta }{\mathrm{d} t}=0-2\frac{c}{2ab}\times \frac{\mathrm{d} c}{\mathrm{d} t}[/tex]
at [tex]\theta =\frac{\pi }{3}[/tex]
c=11.53 m
substituting values we get
[tex]\frac{\sqrt{3}}{2}\times 0.02=\frac{11.53}{9\times 13}\times \frac{\mathrm{d} c}{\mathrm{d} t}[/tex]
[tex]\frac{\mathrm{d} c}{\mathrm{d} t}=0.175 m/s[/tex]
The value of dc/dt is given as 0.175m/s
Given a triangle with sides a, b and c, the cosine rule is expressed as:
[tex]c^2=a^2+b^2-2abcos\theta\\cos\theta=\frac{a^2+b^2-c^2}{2ab}\\cos\theta=\frac{a^2+b^2}{2ab} -\frac{c^2}{2ab}[/tex]
Differentiate both sides of the equation implicitly to have:
[tex]-sin \theta \times \frac{d \theta}{dt}=-\frac{c}{ab}\frac{dc}{dt}[/tex]
Given the following parameters;
[tex]\theta = \frac{\pi}{3}\\ c=11.53 cm[/tex]
a = 9cm
b = 13cm
Substitute the given parameters into the formula to get dc/dt:
[tex]-sin \frac{\pi}{3} \times \frac{d \theta}{dt}=-\frac{c}{ab}\frac{dc}{dt}\\-\frac{\sqrt{3}}{2} \times 0.02=-\frac{11.53}{9\times 13}\frac{dc}{dt}[/tex]
On solving, the value of dc/dt is given as 0.175m/s
Learn more on the rate of change here: https://brainly.com/question/8728504
