A-level · Differentiation

Connected rates of changeExam-style questions

Working out how fast one quantity changes from how fast another does. Why linked quantities have linked rates; the chain rule as the bridge, dV/dt = (dV/dh)(dh/dt); finding the formula that links them in one variable; differentiating before substituting; signs and units for quantities that shrink; and chains of three rates, including a point moving along a curve.

Start practising Read the lesson

22 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4

1
Water drains from an inverted cone whose height always equals its radius, so $V = \tfrac13\pi r^3$ , at $8\text{ cm}^3\text{s}^{-1}$ . Find the rate at which the radius is decreasing when $r = 4\text{ cm}$ .
Level 3 · 3 marksTry it
2
A cylindrical rainwater tank has a radius of $0.8$ $0.8$ m and is being filled with water from a downpipe.

In a model for the filling of the tank,
- the walls of the tank are treated as having no thickness
- the volume of water in the tank increases at a rate that is inversely proportional to the depth of the water
The depth of the water in the tank is $h$ $h$ metres, $t$ $t$ minutes after the tank starts to fill.
(a) Show that

$\frac{dh}{dt} = \frac{\lambda}{h}$

where $\lambda$ $λ$ is a positive constant.

(3 marks)

When the depth of the water is $0.5$ $0.5$ m, the depth is increasing at a rate of $0.04$ $0.04$ m per minute.
(b) Write down the value of $\lambda$ $λ$ .

(1 mark)
Level 3 · 4 marksTry it
3
A spherical balloon is inflated at $100\text{ cm}^3\text{s}^{-1}$ . Show that its surface area $S = 4\pi r^2$ increases at $\dfrac{200}{r}\text{ cm}^2\text{s}^{-1}$ , and find this rate when $r = 5\text{ cm}$ .
Level 3 · 4 marksTry it
4
A soap bubble is modelled as a sphere of radius $r$ cm. Air is blown into the bubble so that its volume, $V$ cm³, increases at a constant rate of $k$ cm³ s⁻¹, where $k$ is a positive constant.

The surface area of the bubble is $S$ cm².

Show that, according to the model, the rate at which $S$ is increasing is inversely proportional to $r$ .

[The volume $V$ and the surface area $S$ of a sphere of radius $r$ are given by $V = \dfrac{4}{3}\pi r^3$ and $S = 4\pi r^2$ ]

(4 marks)
Level 3 · 4 marksTry it
5
Air is pumped into a spherical balloon at $50\text{ cm}^3\text{s}^{-1}$ . Find the rate at which the radius is increasing when $r = 5\text{ cm}$ . $\left(V = \tfrac43\pi r^3\right)$
Level 3 · 3 marksTry it

7 more questions on connected rates of change

Free with an account — just an email, no card. The whole bank, every mark scheme and worked solution.

Start free