$A$ kind of bacteria grows by $t^3$ in $t \ s$. The time taken for the rate of growth of the bacteria to become $1200 \ \text{per } s$ is: (in $s$)

$A$ ladder $5 \ m$ long is leaning against a wall. The bottom of the ladder is pulled along the ground,away from the wall,at the rate of $2 \ cm/s$. How fast is its height on the wall decreasing when the foot of the ladder is $4 \ m$ away from the wall?

The displacement $S$ of a moving particle at a time $t$ is given by $S=5+\frac{48}{t}+t^3$. Then its acceleration when the velocity is zero,is

$A$ is a point on the circle with radius $8$ and centre at $O$. $A$ particle $P$ is moving on the circumference of the circle starting from $A$. $M$ is the foot of the perpendicular from $P$ on $OA$ and $\angle POM = \theta$. When $OM = 4$ and $\frac{d\theta}{dt} = 6 \text{ radians/sec}$,then the rate of change of $PM$ is (in units/sec)

$A$ particle is moving in a straight line according to the equation $s = 45t + 11t^2 - t^3$. The time at which it will come to rest is ......... $sec$.

$A$ water tank has the shape of an inverted right circular cone with its axis vertical and vertex lowermost. Its semi-vertical angle is $\tan ^{-1}(0.5)$. Water is poured into it at a constant rate of $5 \ m^3/h$. Find the rate at which the level of the water is rising at the instant when the depth of the water in the tank is $4 \ m$.