An object is moving in the clockwise direction around the unit circle x2+y2=1. As it passes through the point (1/2,3/2), its y-coordinate is decreasing at the rate of 3 units per second. The rate at which the x-coordinate changes at this point is (in units per second)

# An object is moving in the clockwise direction around the unit circle ${x}^{2}+{y}^{2}=1$. As it passes through the point $\left(1/2,\sqrt{3}/2\right)$, its y-coordinate is decreasing at the rate of 3 units per second. The rate at which the x-coordinate changes at this point is (in units per second)

1. A

2

2. B

$3\sqrt{3}$

3. C

$\sqrt{3}$

4. D

$2\sqrt{3}$

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### Solution:

We find $\frac{dx}{dt}$ when $x=\frac{1}{2}$ and $y=\frac{\sqrt{3}}{2}$ given

that $\frac{dy}{dt}=-3$ units/s and  ${x}^{2}+{y}^{2}=1$.

Differentiating ${x}^{2}+{y}^{2}=1$, we have

$2x\frac{dx}{dt}+2y\frac{dy}{dt}=0.$

Putting $x=1/2,y=\sqrt{3}/2$ and ,we have

$\frac{1}{2}\frac{dx}{dt}+\frac{\sqrt{3}}{2}\left(-3\right)=0⇒\frac{dx}{dt}=3\sqrt{3}$.

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