The side of an equilateral triangle expands at the rate of $2 \, cm/s$. The rate of increase of its area when each side is $10 \, cm$ is:

$A$ $10 \, cm$ long rod $AB$ moves with its ends on two mutually perpendicular straight lines $OX$ and $OY$. If the end $A$ is moving at the rate of $2 \, cm/sec$,then when the distance of $A$ from $O$ is $8 \, cm$,the rate at which the end $B$ is moving is:

$A$ bullet is shot horizontally and its distance $S$ cm at time $t$ second is given by $S=1200t-15t^2$. Then, the distance covered by the bullet when it comes to rest is: (in $\text{ cm}$)

$A$ spherical balloon is filled with $4500\pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72\pi$ cubic meters per minute,then the rate (in meters per minute) at which the radius of the balloon decreases $49$ minutes after the leakage began is:

$A$ street light is at the top of a $12 \ m$ pole. $A$ man $2 \ m$ tall walks away from the pole towards a wall $12 \ m$ away from the pole at a speed of $1/2 \ m/s$. The rate at which his shadow on the wall is decreasing when he is $8 \ m$ from the wall is:

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 $y$-coordinate is decreasing at the rate of $3 \text{ units/sec}$. The rate at which the $x$-coordinate changes at this point is