$A$ point is moving on the curve $y=x^3-3x^2+2x-1$ and the $y$-coordinate of the point is increasing at the rate of $6 \text{ units/sec}$. When the point is at $(2, -1)$,the rate of change of the $x$-coordinate of the point is:

$A$ particle moves along the curve $y = x^{3/2}$ in the first quadrant in such a way that its distance from the origin increases at the rate of $11$ units per second. The value of $\frac{dx}{dt}$ when $x = 3$ is

$A$ sphere increases its volume at the rate of $\pi \text{ cm}^3/\text{s}$. The rate at which its surface area increases when the radius is $1 \text{ cm}$ is

An ice layer of uniform thickness is deposited on an iron sphere of radius $10 \text{ cm}$. The ice melts at a rate of $50 \text{ cm}^3/\text{min}$. When the thickness of the ice layer is $5 \text{ cm}$,the rate at which the thickness of the ice layer is decreasing is ...... $\text{cm/min}$.

$A$ man of height $2 \ m$ walks at a uniform speed of $5 \ km/hr$ away from a lamp post which is $6 \ m$ high. Find the rate at which the length of his shadow increases in $km/hr$.

$A$ 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 $15 \ cm$.