Mathematics, 18.12.2019 10:31 maddie953
Need to know the formula to use and how to work the implicit differentiation afterwards! a particle on the x-axis is moving to the right at 2 units per second. a second particle is moving down the y-axis at the rate of 3 units per second. at a certain instant the first particle is at the point (5,0) and the second is at the point (0,7). how rapidly is the angle between the x-axis and the line joining the two particles changing at that instant? are the particles moving towards or away from each other at that instant?
Answers: 3
![\bf tan(\theta)=\cfrac{y}{x}\impliedby \textit{using the quotient rule} \\\\\\ sec^2(\theta)\cfrac{d\theta}{dt}=\cfrac{\frac{dy}{dt}x-y\frac{dx}{dt}}{x^2}\implies \cfrac{1}{cos^2(\theta)}\cdot \cfrac{d\theta}{dt}=\cfrac{\frac{dy}{dt}x-y\frac{dx}{dt}}{x^2} \\\\\\ \cfrac{d\theta}{dt}=cos^2(\theta)\cdot \cfrac{\frac{dy}{dt}x-y\frac{dx}{dt}}{x^2}\qquad \begin{cases} \cfrac{dy}{dt}=3\\ \cfrac{dx}{dt}=2\\ x=5\\ y=7\\ [cos(\theta)]^2=\left( \frac{5}{\sqrt{74}} \right)^2=\frac{25}{74} \end{cases} \\\\\\](/tpl/images/0424/1281/1d931.png)
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Need to know the formula to use and how to work the implicit differentiation afterwards! a particle...
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