Hello, fxsapa!
We have: point O, anywhere on the plane; line AB, anywhere on the plane.
Point M, moving on AB from A, at speed described by function s(t).
Find the rate of change of 
(from time t),
given that you can know everything about 
(lengths, angles, ...)
but that this triangle is scalene. Code:
O
*
*/ *
* / *
b * θ/ *
* / *
* / *
* / *
* φ / *
A * * * * * * * * * * * B
x M
We have: . 
. . . .and: . 
In 
, apply the Law of Sines:
. . ![\frac{\sin\theta}{x} \:=\:\frac{\sin(180-A-\theta)}{b} \quad\Rightarrow\quad b\sin\theta \;=\;x\sin([180 - A] - \theta)<br />](http://latex.codecogs.com/png.latex?\frac{\sin\theta}{x} \:=\:\frac{\sin(180-A-\theta)}{b} \quad\Rightarrow\quad b\sin\theta \;=\;x\sin([180 - A] - \theta)<br />
)
. . ![b\sin\theta \;=\;x\bigg[\underbrace{\sin(180-A)}_{\text{This is }\sin A}\cos\theta - \underbrace{\cos(180-A)}_{\text{This is }-\cos A}\sin\theta\bigg]](http://latex.codecogs.com/png.latex?b\sin\theta \;=\;x\bigg[\underbrace{\sin(180-A)}_{\text{This is }\sin A}\cos\theta - \underbrace{\cos(180-A)}_{\text{This is }-\cos A}\sin\theta\bigg] )
. . ![b\sin\theta \;=\;x\underbrace{\bigg[\sin A\cos\theta + \cos A\sin\theta\bigg]}_{\text{This is }\sin(\theta + A)}](http://latex.codecogs.com/png.latex?b\sin\theta \;=\;x\underbrace{\bigg[\sin A\cos\theta + \cos A\sin\theta\bigg]}_{\text{This is }\sin(\theta + A)} )
. . )
Differentiate with respect to time:
. . \,\frac{d\theta}{dt} + \sin(\theta + A)\,\frac{dx}{dt})
. . \,\frac{d\theta}{dt} \;=\;\sin(\theta + A)\,\frac{dx}{dt})
. . ![\bigg[b\cos\theta - x\cos(\theta+A)\bigg]\,\frac{d\theta}{dt} \;=\;\sin(\theta + A)\,\frac{dx}{dt}](http://latex.codecogs.com/png.latex?\bigg[b\cos\theta - x\cos(\theta+A)\bigg]\,\frac{d\theta}{dt} \;=\;\sin(\theta + A)\,\frac{dx}{dt})
Therefore: . \,\frac{dx}{dt}}{b\cos\th eta - x\cos(\theta +A)} )
LinkBack URL
About LinkBacks
