At present,a firm is manufacturing $1000$ 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 $9$ more workers,then the new level of production of items is:

Coffee is draining from a conical filter,with both height and diameter equal to $15 \, cm$,into a cylindrical coffee pot with a diameter of $15 \, cm$. The rate at which coffee drains from the filter into the pot is $100 \, cm^3/min$. The rate in $cm/min$ at which the level in the pot is rising at the instant when the coffee in the pot is $10 \, cm$ deep,is:

$A$ bulb is placed at the centre of a circular track of radius $10 \ m$. $A$ vertical wall is erected touching the track at a point $P$. $A$ man is running along the track with a speed of $10 \ m/sec$. Starting from $P$, the speed with which his shadow is running along the wall when he is at an angular distance of $60^{\circ}$ from $P$ is: (in $m/sec$)

$A$ 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

Moving along the $x$-axis are two points with $x = 10 + 6t$ and $x = 3 + t^2$. The speed with which they are receding from each other at the time of encounter is ........... $cm/sec$. ($x$ is in $cm$ and $t$ is in seconds)

Water is being poured into a cylindrical vessel with a radius of $2 \text{ cm}$ at a rate of $8 \text{ cm}^3/\text{s}$. The rate at which the height of the water is increasing is: