The sides of an equilateral triangle are increasing at the rate of $2 \, cm/sec$. The rate at which the area increases,when the side is $10 \, cm$ is:

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

The vertical angle of a right circular cone is $60^{\circ}$. If water is being poured into the cone at the rate of $\frac{1}{\sqrt{3}} \text{ m}^3/\text{min}$,then the rate $(\text{m/min})$ at which the radius of the water level is increasing when the height of the water level is $3 \text{ m}$ is

Gas is being pumped into a spherical balloon at the rate of $30 \ ft^3 / \text{min}$. The rate at which the radius increases when it reaches the value $15 \ ft$,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.

The distance $s$ in meters travelled by a particle in $t$ seconds is given by $s = \frac{2 t^3}{3} - 18 t + \frac{5}{3}$. The acceleration when the particle comes to rest is (in $m/s^2$)