$A$ ladder,$5 \ m$ long,rests against a vertical wall. If its top slides downwards at the rate of $10 \ cm/s$,then the angle between the ladder and the floor is decreasing at the rate of how many radians/second when its lower end is $4 \ m$ away from the wall?

$A$ spherical balloon is filled with $4500\pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72\pi$ cubic meters per minute,then the rate (in meters per minute) at which the radius of the balloon decreases $49$ minutes after the leakage began is:

Find the point on the parabola $y^2 = 18x$ where the rate of change of the $y$-coordinate is twice the rate of change of the $x$-coordinate.

$A$ man of $5 \text{ feet}$ height is walking away from a light fixed at a height of $15 \text{ feet}$ at the rate of $K \text{ miles/hour}$. If the rate of increase of his shadow is $\frac{11}{5} \text{ feet/sec}$, then $K=$ (Take $1 \text{ mile} = 5280 \text{ feet}$)

The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x) = 3x^2 + 36x + 5$. Find the marginal revenue when $x = 5$,where marginal revenue is defined as the rate of change of total revenue with respect to the number of items sold at an instant.

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)