Paint is poured onto a flat surface and a circular patch is formed. The area of the patch increases at a rate of $5 {\mathrm{cm}}^2/\mathrm{s}$. Find, in terms of $\mathrm{\pi }$, the radius of the patch after $8$ seconds.

Paint is poured onto a flat surface and a circular patch is formed. The area of the patch increases at a rate of $5 {\mathrm{cm}}^2/\mathrm{s}$. Find, in terms of $\mathrm{\pi }$, the rate of increase of the radius of the patch after $8$ seconds.

A cylindrical container of radius $8$ cm and height $25$ cm is completely filled with water. The water is then poured at a constant rate from the cylinder into an empty inverted cone.The cone has radius $15$ cm and height $24$ cm and its axis is vertical. It takes $40$  seconds for all of the water to be transferred. If $V$ represents the volume of water, in ${\mathrm{cm}}^3$, in the cone at time $t$ seconds, find $\frac{dV}{dt}$ in terms of $\mathrm{\pi }$.

A cylindrical container of radius $8$ cm and height $25$ cm is completely filled with water. The water is then poured at a constant rate from the cylinder into an empty inverted cone.The cone has radius $15$ cm and height $24$ cm and its axis is vertical. It takes $40$  seconds for all of the water to be transferred. When the depth of the water in the cone is $10$ cm, find the rate of change of the height of the water in the cone.

A cylindrical container of radius $8$ cm and height $25$ cm is completely filled with water. The water is then poured at a constant rate from the cylinder into an empty inverted cone.The cone has radius $15$ cm and height $24$ cm and its axis is vertical. It takes $40$  seconds for all of the water to be transferred. When the depth of the water in the cone is $10$ cm, Find the rate of change of the horizontal surface area of the water in the cone.

The volume of a spherical balloon is increasing at a constant rate of $40 {\mathrm{cm}}^3/\mathrm{s}$ per second. Find the rate of increase of the radius of the balloon when the radius is $15$ cm.

An oil pipeline under the sea is leaking oil and a circular patch of oil has formed on the surface of the sea. At midday the radius of the patch of oil is $50$ m and is increasing at a rate of $3$ metres per hour. Find the rate at which the area of the oil is increasing at midday.

A curve has equation; $y=27x-\frac{4}{(x+2{)}^2}$. Show that the curve has a stationary point at $x=-\frac{8}{3}$ and determine its nature.

A watermelon is assumed to be spherical in shape while it is growing. Its mass, $M$ kg, and radius, $r$ cm, are related by the formula $M=k{r}^3$, where $k$ is a constant. It is also assumed that the radius is increasing at a constant rate of $0.1$ centimetres per day. On a particular day the radius is $10$ cm and the mass is $3.2$ kg. Find the value of $k$ and the rate at which the mass is increasing on this day.