$x$ and $y$ are the sides of two squares such that $y = x - x^{2}$. Find the rate of change of the area of the second square with respect to the area of the first square.

$OA$ and $OB$ are two roads enclosing an angle of $120^{\circ}$. $X$ and $Y$ start from '$O$' at the same time. $X$ travels along $OA$ with a speed of $4 \text{ km/h}$ and $Y$ travels along $OB$ with a speed of $3 \text{ km/h}$. The rate at which the shortest distance between $X$ and $Y$ is increasing after $1 \text{ h}$ is

The $x$-coordinate changes on the curve $y=3x^5+15x-8$ at the rate of $\frac{1}{5} \text{ units/sec}$. If $A(x_1, y_1)$ and $B(x_2, y_2)$ are the points on the curve at which the $y$-coordinate changes at the rate of $6 \text{ units/sec}$, then the slope of $AB$ is:

If the volume of a sphere increases at the rate of $2 \pi \text{ cm}^3/\text{s}$,then the rate of increase of its radius (in $\text{cm}/\text{s}$),when the volume is $288 \pi \text{ cm}^3$,is

The distance $s$ in meters covered by a particle in $t$ seconds is given by $s = 2 + 27t - t^3$. The particle will stop after covering a distance of:

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 $g = 32 \ ft/sec^2$)