If the rate of change of the slope of the tangent drawn to the curve $y=x^3-2x^2+3x-2$ at the point $(2,4)$ is $k$ times the rate of change of its abscissa, then $k=$

From a balloon rising vertically with a uniform velocity of $v \ ft/sec$, a stone is dropped. If the stone reaches the ground after $4 \ sec$, what is the height of the balloon above the ground at that moment (in $ft$)? (Take $g = 32 \ ft/sec^2$)

$A$ stone is dropped into a quiet lake and waves move in circles at the speed of $5 \text{ cm/s}$. At the instant when the radius of the circular wave is $8 \text{ cm}$,how fast is the enclosed area increasing?

$A$ square plate is contracting at the uniform rate $4 \,cm^2 / sec$. The rate at which the perimeter is decreasing, when the side of the square is $20 \,cm$, is

The radius of a cylinder is increasing at the rate of $3 \, m/sec$ and its altitude is decreasing at the rate of $4 \, m/sec$. The rate of change of volume when the radius is $4 \, m$ and the altitude is $6 \, m$ is:

The total cost $C(x)$ in Rupees,associated with the production of $x$ units of an item is given by $C(x) = 0.005x^{3} - 0.02x^{2} + 30x + 5000$. Find the marginal cost when $3$ units are produced,where marginal cost is defined as the instantaneous rate of change of total cost with respect to the number of units produced.