$A$ stone is dropped into a quiet lake and waves move in circles at a speed of $8 \,cm/sec$. At the instant when the radius of the circular wave is $12 \,cm$, how fast is the enclosed area increasing?

Consider a rectangle whose length is increasing at the uniform rate of $2 \, m/sec$,breadth is decreasing at the uniform rate of $3 \, m/sec$ and the area is decreasing at the uniform rate of $5 \, m^2/sec$. If after some time the breadth of the rectangle is $2 \, m$,then the length of the rectangle is ........ $m$.

The volume of a spherical balloon is increasing at the rate of $2 \ cm^3/sec$. When its radius is $4 \ cm$, the rate of change of its surface area (in $cm^2/sec$) is

The equation of motion of a particle is given by $s = 2t^3 - 9t^2 + 12t + 1$,where $s$ and $t$ are measured in $cm$ and $sec$. The time when the particle stops momentarily is

The point on the curve $6y = x^3 + 2$ at which the $y$-coordinate is changing $8$ times as fast as the $x$-coordinate is:

The length $x$ of a rectangle is decreasing at the rate of $5 \text{ cm/min}$ and the width $y$ is increasing at the rate of $4 \text{ cm/min}$. When $x = 8 \text{ cm}$ and $y = 6 \text{ cm}$,find the rate of change of the perimeter.