$A$ firm is manufacturing $2000$ items. It is estimated that the rate of change of production $P$ with respect to the additional number of workers $x$ is given by $\frac{dP}{dx} = 100 - 12\sqrt{x}$. If the firm employs $25$ more workers,then the new level of production of items is:

If $V = \frac{4}{3}\pi r^3$,find the rate of change of $V$ with respect to $t$ when $r = 10$ and $\frac{dr}{dt} = 0.01$.

Water is dripping out at a steady rate of $1 \text{ cm}^3/\text{sec}$ through a tiny hole at the vertex of a conical vessel,whose axis is vertical. When the slant height of water in the vessel is $4 \text{ cm}$,find the rate of decrease of the slant height,where the semi-vertical angle of the conical vessel is $\frac{\pi}{6}$.

$A$ 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:

The displacement of a particle in time $t$ is given by $s = 2t^2 - 3t + 1$. The acceleration is

If the distance travelled by a point in time $t$ is $s = 180t - 16t^2$,then the rate of change in velocity is ......... $unit$.