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.

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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
A spherical balloon is being inflated so that its volume increases at the constant rate of $200$ cm³ s⁻¹.
(a) Find the rate at which the radius is increasing at the instant when the radius is $5$ cm, giving your answer exactly. (4 marks)
(b) Find the rate at which the surface area is increasing at the same instant. (3 marks)

[The volume and surface area of a sphere are $V = \frac{4}{3}\pi r^3$ and $S = 4\pi r^2$ .]
Level 4 · 7 marksTry it
2
Sand is poured onto level ground at a constant rate of $0.5$ m $^3$ per minute. The sand forms a pile in the shape of a right circular cone whose height $h$ metres is always equal to twice the base radius.
(a) Show that the volume $V$ m $^3$ of the pile is given by $V = \dfrac{\pi h^3}{12}$ . (2 marks)
(b) Determine the exact rate at which the height of the pile is increasing at the instant when $h = 3$ . (4 marks)
(c) Find the rate at which the base radius is increasing at the same instant. (2 marks)
Level 4 · 8 marksTry it
3
Figure 5 shows a container in the shape of a right circular cone with its vertex pointing downwards. The cone has radius $6$ cm and height $18$ cm.Figure 5 (not accurately drawn)

Water is poured into the container at a constant rate of $8$ cm³ s⁻¹. At time $t$ seconds the depth of the water is $h$ cm and the radius of the water surface is $r$ cm.
(a) Show that the volume $V$ cm³ of water in the container is given by $V = \dfrac{\pi h^3}{27}$ . (3 marks)
(b) Find the rate at which the depth of the water is increasing at the instant when $h = 6$ , giving your answer to $3$ significant figures. (4 marks)
(c) Find the rate at which the area of the water surface is increasing at that same instant, giving your answer to $3$ significant figures. (3 marks)

[The volume of a cone of radius $r$ and height $h$ is $\tfrac13\pi r^2 h$ .]
Level 4 · 10 marksTry it
4
A curve has equation $y = \dfrac{6}{5 - 3x}$ .
(a) Find $\dfrac{dy}{dx}$ .

(2 marks)

A point $P$ moves along the curve. At a particular instant, the $x$ -coordinate of $P$ is increasing at a rate of $0.04$ units per second and the $y$ -coordinate of $P$ is increasing at a rate of $0.18$ units per second.
(b) Find the possible $x$ -coordinates of $P$ at this instant.

(4 marks)
Level 4 · 6 marksTry it
5
A cube is expanding. At time $t$ seconds the length of each edge is $x$ cm, the volume is $V$ cm³ and the total surface area is $A$ cm².

The volume is increasing at the constant rate of $24$ cm³ s⁻¹.
(a) Find the rate at which $x$ is increasing at the instant when $x = 4$ . (4 marks)
(b) Find the rate at which $A$ is increasing at that same instant. (3 marks)
(c) Show that, at any instant, the rate at which the surface area is increasing is inversely proportional to $x$ . (2 marks)
Level 4 · 9 marksTry it

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