Answer:
The rate of angle is 26.25 rad/sec.
Explanation:
Given that,
First side of triangle a= 20 cm
Second side of triangle b= 50 cm
One side of a triangle is increasing at a rate = 5 cm/sec
Second side is increasing at a rate = 7 cm/s
Angle [tex]\theta=\dfrac{\pi}{3}=60^{\circ}[/tex]
If the area of the triangle remains constant,
We need to calculate rate of angle
Using formula of area of triangle
[tex]A=\dfrac{1}{2}ab\sin\theta[/tex]
On differentiating
[tex]\dfrac{dA}{dt}=\dfrac{1}{2}ab\cos\theta\dfrac{d\theta}{dt}+\dfrac{1}{2}a\sin\theta\dfrac{db}{dt}+\dfrac{1}{2}b\sin\theta\dfrac{da}{dt}[/tex]
Put the value into the formula
[tex]0=\dfrac{1}{2}\times20\times50\cos60\dfrac{d\theta}{dt}+\dfrac{1}{2}\times20\sin60\times7+\dfrac{1}{2}\times50\sin60\times5[/tex]
[tex]\dfrac{d\theta}{dt}=\dfrac{35\sqrt{3}\times\dfrac{25}{2}\sqrt{3}\times5}{250}[/tex]
[tex]\dfrac{d\theta}{dt}=26.25\ rad/sec[/tex]
Hence, The rate of angle is 26.25 rad/sec.
