Related rates of change: notes and practice questions

A designer is creating a decorative glass container shaped like a dome. The outer profile of the container can be modelled by the function $f(x) = \sqrt{9-x^2}$, where $0 \le x \le 3$ and $x$ and $y$ are measured in metres.

Sketch the curve $y = f(x)$, clearly indicating the coordinates of the endpoints.

Show that the inverse function of $f$ is given by $f^{-1}(x) = \sqrt{9-x^2}$.

State the domain and range of $f^{-1}$.

The container is formed by rotating the curve $y = f(x)$ by $2\pi$ about the y-axis. Show that the volume, $V \text{ m}^3$, of liquid in the container when it is filled to a height of $h$ metres is given by $V = \pi \left( 9h - \frac{1}{3}h^3 \right)$.

Hence, determine the maximum volume of the container.

At $t = 0$, the container is empty. Liquid is then added to the container at a constant rate of $0.5 \text{ m}^3\text{s}^{-1}$.

Find the time it takes to fill the container to its maximum volume.

Find the rate of change of the height of the liquid when the container is filled to half its maximum volume.

If a balloon that has a shape of a sphere is being inflated with helium at a rate of 0.2 $m^{3}s^{- 1}$.

$a$ Find the radius of the balloon at which the volume according to the given model would be $2\pi$ $m^{3}$.

$b$ Calculate the rate of change of the radius of this model for the calculated value in the previous part.

A machine pours sand onto a flat horizontal surface at a constant rate of 0.5 m³ per minute. The sand forms a pile in the shape of a right circular cone whose base radius is always equal to its height.

(a) Find the height of the pile when the volume of sand is 12 m³.

(b) Find the rate at which the height of the pile is increasing at the instant when the volume of sand is 12 m³.

A machine is pouring gravel onto a pile, which forms the shape of a right circular cone. The height of the cone, $h$, is always equal to the diameter of its base. The gravel is being added at a constant rate of 1.5 $m^{3}min^{-1}$.

(a) Find the height of the conical pile when its volume is 10 $m^{3}$.

(b) Calculate the rate at which the height of the pile is increasing at the instant when the volume is 10 $m^{3}$.

A spherical hailstone melts such that its volume decreases at a constant rate of 12 mm³s⁻¹.

(a) Find the radius of the hailstone, in mm, when its volume is $972\pi$ mm³.

(b) Calculate the rate of change of the radius of the hailstone at the instant its volume is $972\pi$ mm³.

A pile of gravel is being formed in the shape of a right circular cone whose base radius, $r$, is always equal to its height, $h$. Gravel is being added to the pile at a rate of $0.5 \, \text{m}^3 \text{min}^{-1}$.

(a) Find the height of the pile when its volume is $12 \, \text{m}^3$.

(b) Calculate the rate at which the height of the pile is increasing at the instant when its volume is $12 \, \text{m}^3$.