Answer :
Answer:
1.692 m/min
Step-by-step explanation:
Let [tex]\theta[/tex] be the angle between the two sides and x be the length of the third side. By cosine rule,
[tex]x^2 = 12^2+14^2-2\times12\times14\cos\theta = 340 - 336\cos\theta[/tex]
[tex]x= \sqrt{340 - 336\cos\theta}[/tex]
We differentiate x with respect to [tex]\theta[/tex] by applying chain rule.
[tex]\dfrac{dx}{d\theta} = \dfrac{336\sin\theta}{2\sqrt{340 - 336\cos\theta}} = \dfrac{168\sin\theta}{\sqrt{340 - 336\cos\theta}}[/tex]
Rate of change of [tex]\theta[/tex] is 2
[tex]\dfrac{\theta}{dt} = 2[/tex]
Rate of change of x is
[tex]\dfrac{dx}{dt} = \dfrac{dx}{d\theta}\times\dfrac{d\theta}{dt}[/tex]
[tex]\dfrac{dx}{dt} = \dfrac{168\sin\theta}{\sqrt{340 - 336\cos\theta}} \times2=\dfrac{336\sin\theta}{\sqrt{340 - 336\cos\theta}}[/tex]
At 60°,
[tex]\dfrac{dx}{dt} = \dfrac{336\sin60}{\sqrt{340 - 336\cos60}} = 1.692 \text{ m/min}[/tex]
