$A$ ladder of length $13 \ m$ has one end resting against a vertical wall and the other on the ground. If the lower end moves away from the wall at a speed of $2 \ m/min$, then the speed (in $m/min$) at which the upper end falls when the bottom is $5 \ m$ away from the wall is

The radius of a right circular cylinder increases at the rate of $0.1 \text{ cm/min}$,and the height decreases at the rate of $0.2 \text{ cm/min}$. The rate of change of the volume of the cylinder,in $\text{cm}^3/\text{min}$,when the radius is $2 \text{ cm}$ and the height is $3 \text{ cm}$ 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)

If the volume of a spherical ball is increasing at the rate of $4 \pi \ cm^3/sec$,then the rate of change of its surface area when the volume is $288 \pi \ cm^3$ is:

$A$ balloon,which always remains spherical,is being inflated by pumping in $900 \ cm^3/sec$ of gas. Find the rate at which the radius of the balloon is increasing when the radius is $15 \ cm$.

$A$ balloon starting from rest is ascending from the ground with a uniform acceleration of $4 \ ft/sec^2$. At the end of $5 \ sec$, a stone is dropped from it. If $T$ is the time taken for the stone to reach the ground and $H$ is the height of the balloon when the stone reaches the ground, then: