If the surface area of a sphere of radius $r$ is increasing uniformly at the rate $8 \, cm^2/s$,then the rate of change of its volume is

$A$ ladder $5 \ m$ long rests against a vertical wall. If its top slides downwards at the rate of $10 \ cm/sec$,then the foot of the ladder is sliding at the rate of . . . . . . $m/sec$,when it is $4 \ m$ away from the wall.

$A$ cube of ice melts without changing its shape at a uniform rate of $4 \, cm^3/min$. The rate of change of the surface area of the cube,in $cm^2/min$,when the volume of the cube is $125 \, cm^3$,is:

$A$ street light is at the top of a $12 \ m$ pole. $A$ man $2 \ m$ tall walks away from the pole towards a wall $12 \ m$ away from the pole at a speed of $1/2 \ m/s$. The rate at which his shadow on the wall is decreasing when he is $8 \ m$ from the wall is:

If a spherical balloon has a variable diameter $d = 3x + \frac{9}{2}$ units, then the rate of change of its volume $V$ with respect to $x$ is:

Let $x$ be the length of one of the equal sides of an isosceles triangle,and let $\theta$ be the angle between them. If $x$ is increasing at the rate of $1/12 \ m/hr$,and $\theta$ is increasing at the rate of $\pi/180 \ \text{radians/hr}$,then find the rate in $m^2/hr$ at which the area of the triangle is increasing when $x = 12 \ m$ and $\theta = \pi/4$.