If the radius of a spherical balloon is increasing at the rate of $5 \text{ inch/min}$, then the rate at which the volume increases (in $\text{cubic inches/min}$) when the radius is $10 \text{ inches}$ is: (in $\pi$)

Let $x$ be the length of one of the equal sides of an isosceles triangle,and let $\theta$ be the angle between them. If $x$ is increasing at the rate of $1/12 \ m/hr$,and $\theta$ is increasing at the rate of $\pi/180 \ \text{radians/hr}$,then find the rate in $m^2/hr$ at which the area of the triangle is increasing when $x = 12 \ m$ and $\theta = \pi/4$.

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:

The radius of a cylinder is increasing at the rate of $5 \ cm/min$,so that its volume remains constant. When its radius is $5 \ cm$ and height is $3 \ cm$,the rate of decrease of its height is .......... $cm/min$.

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.

After $t$ seconds,the acceleration of a particle,which starts from rest and moves in a straight line is $(8-\frac{t}{5}) \text{ cm/s}^2$. The velocity of the particle at the instant when the acceleration is zero is: (in $\text{ cm/s}$)