An object is moving in the clockwise direction around the unit circle $x^2+y^2=1$. As it passes through the…
An object is moving in the clockwise direction around the unit circle $x^2+y^2=1$. As it passes through the point $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$, its $y$-co-ordinate is decreasing at the rate of 3 units per sec. The rate at which the $x$-co-ordinate changes at this point is
$2$ units $/ \mathrm{sec}$
$3 \sqrt{3}$ units $/ \mathrm{sec}$
$\sqrt 3$ units $/ \mathrm{sec}$
$2\sqrt 3$ units $/ \mathrm{sec}$
Solution
Given equation is $x^2+y^2=1$ Differentiating w.r.t. t, we get
$\begin{aligned}
& 2 x \frac{\mathrm{d} x}{\mathrm{dt}}+2 y \frac{\mathrm{d} y}{\mathrm{dt}}=0 \\
& 2 x \frac{\mathrm{d} x}{\mathrm{dt}}+2 y(-3)=0 \\
& \frac{\mathrm{d} x}{\mathrm{dt}}=\frac{6 y}{2 x} \\
& \frac{\mathrm{d} x}{\mathrm{dt}}=\frac{3 y}{x} \\
& \left.\frac{\mathrm{d} x}{\mathrm{dt}}\right|_{\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)}=\frac{3 \times \frac{\sqrt{3}}{2}}{\frac{1}{2}} \\
& =3 \sqrt{3} \text { units/sec }
\end{aligned}$