The volume of a spherical balloon is increasing at the rate of $30 \ cm^3/min$. Find the rate of change of the surface area of the balloon when its radius is $6 \ cm$.

If the area of a circle increases at the rate of $\frac{1}{\sqrt{\pi}}$ sq. units/sec,then the rate (in units/sec) at which the perimeter of the circle changes,when perimeter is $\sqrt{\pi}$ units,is

The weight $W$ of a certain stock of fish is given by $W = nw,$ where $n$ is the size of the stock and $w$ is the average weight of a fish. If $n$ and $w$ change with time $t$ as $n = 2t^2 + 3$ and $w = t^2 - t + 2,$ then the rate of change of $W$ with respect to $t$ at $t = 1$ is

If the displacement $S$ of a particle travelling along a straight line in $t$ seconds is given by $S = 2t^3 + 2t^2 - 2t - 3$,then the time taken (in seconds) by the particle to change its direction is

The side of a square sheet of metal is increasing at the rate of $3 \text{ cm/min}$. At what rate is the area increasing when the length of the side is $6 \text{ cm}$?

If the distance $s$ traveled by a particle in time $t$ is $s = a \sin t + b \cos 2t$,then the acceleration at $t = 0$ is