If displacement $s = 5 \sin(2t)$,then the velocity at the end of $t = \frac{\pi}{3} \text{ s}$ is

$A$ vessel in the shape of an inverted cone of height $10 \ ft$ and semi-vertical angle $30^{\circ}$ is full of water. Due to a hole at the vertex, the slant height of the water in the vessel is decreasing at a constant rate of $\frac{1}{\sqrt{3}} \ ft/min$. The rate (in $cu. \ ft/min$) at which the volume of water in the vessel is decreasing, when the volume of water is $\frac{8 \pi}{\sqrt{3}} \ cu. \ ft$, is

$A$ particle moves in a straight line such that $s = \sqrt{t}$,then its acceleration is proportional to

The radius of a circular plate is increasing at the rate of $0.01 \text{ cm/sec}$. When the radius is $12 \text{ cm}$,the rate at which the area increases is:

$A$ car starts from a point $P$ at time $t=0$ seconds and stops at point $Q$. The distance $x$,in metres,covered by it in $t$ seconds is given by $x=t^{2}(2-\frac{t}{3})$. Find the time taken by it to reach $Q$ and also find the distance between $P$ and $Q$.

$x$ and $y$ are the sides of two squares such that $y = x - x^{2}$. Find the rate of change of the area of the second square with respect to the area of the first square.