The rate of change of volume of a sphere with respect to its surface area when the radius is $4 \text{ cm}$ is

The weight $W$ of a certain stock of fish is given by $W = nw,$ where $n$ is the size of the stock and $w$ is the average weight of a fish. If $n$ and $w$ change with time $t$ as $n = 2t^2 + 3$ and $w = t^2 - t + 2,$ then the rate of change of $W$ with respect to $t$ at $t = 1$ is

The radius of a circle is increasing at the rate of $2 \text{ cm/sec}$. The rate at which its area is increasing when the radius of the circle is $5 \text{ decimeters}$ is:

$A$ street light is at the top of a $12 \ m$ pole. $A$ man $2 \ m$ tall walks away from the pole towards a wall $12 \ m$ away from the pole at a speed of $1/2 \ m/s$. The rate at which his shadow on the wall is decreasing when he is $8 \ m$ from the wall is:

Let $S$ be the focus of $y^2 = 4x$ and a point $P$ is moving on the curve such that its abscissa is increasing at the rate of $4 \text{ units/sec}$. Then the rate of increase of the projection of $SP$ on the line $x + y = 1$ when $P$ is at $(4, 4)$ is:

$A$ particle is moving in a straight line. At time $t$, the distance of the particle from its starting point is given by $x = t^3 - 6t^2 + t$. Its acceleration will be zero at