The diagram shows a water container in the shape of a triangular prism of length $120$ cm.The vertical cross-section is an equilateral triangle. Water is poured into the container at a rate of $24 {\mathrm{cm}}^3/\mathrm{s}$. Show that the volume of water in the container, $V {\text{cm}}^3$, is given by $V=40\sqrt{3}{h}^2$, where $h$ cm is the height of the water in the container.

The diagram shows a water container in the shape of a triangular prism of length $120$ cm.The vertical cross-section is an equilateral triangle. Water is poured into the container at a rate of $24 {\mathrm{cm}}^3/\mathrm{s}$. Find the rate of change of $h$ when $h=12$ cm.

Water is poured into the hemispherical bowl of radius $5$ cm at a rate of $3\pi {\mathrm{cm}}^3/\mathrm{s}$. After $t$ seconds, the volume of water in the bowl, $V {\mathrm{cm}}^3$ is given by $V=5\pi h^2-\frac{1}{3}\pi h^3$, where $h$ cm is the height of the water in the bowl. Find the rate of change of $h$ when $h=1$ cm.

Water is poured into the hemispherical bowl of radius $5$ cm at a rate of $3\pi {\mathrm{cm}}^3/\mathrm{s}$. After $t$ seconds, the volume of water in the bowl, $V {\mathrm{cm}}^3$ is given by $V=5\pi h^2-\frac{1}{3}\pi h^3$, where $h$ cm is the height of the water in the bowl. Find the rate of change of $h$ when $h=3$ cm.

The diagram shows a right circular cone with radius $10$ cm and height $30$ cm. The cone is initially completely filled with water. Water leaks out of the cone through a small hole at the vertex at a rate of $4 {\mathrm{cm}}^3/\mathrm{s}$. Show that the volume of water in the cone, $V {\mathrm{cm}}^3$, when the height of the water is $h$ cm is given by the formula $V=\frac{\pi h^3}{27}$.

The diagram shows a right circular cone with radius $10$ cm and height $30$ cm. The cone is initially completely filled with water. Water leaks out of the cone through a small hole at the vertex at a rate of $4 {\mathrm{cm}}^3/\mathrm{s}$. Find the rate of change of $h$, when $h=20$ cm.

Oil is poured onto a flat surface and a circular patch is formed. The radius of the patch increases at a rate of $2\sqrt{r} \mathrm{cm}/\mathrm{s}$. Find the rate at which the area is increasing when the circumference is $8\pi \mathrm{cm}$.

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.