Related-rate problems | Year 12

01 / LEARN

The idea, step by step

Write a relation before differentiating

A related-rates problem links two or more quantities that change with time. Introduce symbols for the changing dimensions and write a valid relation, such as $V = \frac{4}{3} π r^{3}$ . Use the same time unit throughout.

Keep changing quantities as variables while differentiating with respect to $t$ . Only then insert the measurements at the requested instant. Replacing a changing radius with a number too early would incorrectly treat it as constant.

Read the sign and units

A positive rate means the chosen quantity is increasing; a negative rate means it is decreasing. If a ladder foot moves away from a wall, the top height decreases, so its vertical rate should be negative.

Check units after solving. A volume rate has cubic units per time; a radius rate has length per time. In a conical container, similar triangles can provide a second relation so that the volume is written in one changing variable.

A common trap

Substitute every measurement before differentiating.

Instead: Differentiate the relation in variables first; substitute the instantaneous values afterwards.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example

A sphere expands with $d r / d t = 0.2$ cm/s. Find $d V / d t$ when $r = 5$ cm.

$V = \frac{4}{3} π r^{3}$ , so $d V / d t = 4 π r^{2} d r / d t$ .
Insert $r = 5$ and $d r / d t = 0.2$ .
$d V / d t = 4 π (25) (0.2) = 20 π$ .

20 π

3

Example 2

Worked example

A 10 m ladder has its foot 6 m from a wall, moving away at 0.5 m/s. Find the top height rate.

Let $x$ be the foot distance and $y$ the height: $x^{2} + y^{2} = 100$ . At $x = 6$ , $y = 8$ .
Differentiate: $2 x \dot{x} + 2 y \dot{y} = 0$ , so $\dot{y} = - x \dot{x} / y$ .
$\dot{y} = - (6) (0.5) / 8 = - 3 / 8$ ; the top moves down.

- 3 / 8

m/s

Check your understanding

If $A = π r^{2}$ , $r = 4$ cm and $d r / d t = 3$ cm/s, find $d A / d t$ .

Show answer

$24 π$ cm $2$ /s

$d A / d t = 2 π r d r / d t = 2 π (4) (3)$ .

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

The side of a square grows at 2 cm/s. Find its area rate when the side is 7 cm.
2 marks
A cube edge grows at 0.4 cm/s. Find its volume rate at edge 3 cm.
2 marks
A circle area shrinks at $12 π$ cm $2$ /s. Find $d r / d t$ at $r = 3$ cm.
2 marks
A 13 m ladder has foot distance 5 m and foot rate 0.6 m/s away from the wall. Find the top rate.
2 marks
Water enters a cylinder of radius 2 m at 3 m $3$ /min. Find the depth rate.
2 marks
A spherical balloon radius is 6 cm when its volume grows at $72 π$ cm $3$ /s. Find its radius rate.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

The side of a square grows at 2 cm/s. Find its area rate when the side is 7 cm.
2 marks
$A = s^{2}$ gives $\dot{A} = 2 s \dot{s}$ .
At $s = 7$ , $\dot{A} = 2 (7) (2) = 28$ .
$28$ cm $2$ /s
A cube edge grows at 0.4 cm/s. Find its volume rate at edge 3 cm.
2 marks
$V = s^{3}$ gives $\dot{V} = 3 s^{2} \dot{s}$ .
$3 (9) (0.4) = 10.8$ .
$10.8$ cm $3$ /s
A circle area shrinks at $12 π$ cm $2$ /s. Find $d r / d t$ at $r = 3$ cm.
2 marks
$\dot{A} = 2 π r \dot{r}$ .
$- 12 π = 6 π \dot{r}$ , so $\dot{r} = - 2$ .
$- 2$ cm/s
A 13 m ladder has foot distance 5 m and foot rate 0.6 m/s away from the wall. Find the top rate.
2 marks
The height is $\sqrt{169 - 25} = 12$ m.
$\dot{y} = - x \dot{x} / y = - 5 (0.6) / 12 = - 0.25$ .
$- 0.25$ m/s
Water enters a cylinder of radius 2 m at 3 m $3$ /min. Find the depth rate.
2 marks
$V = 4 π h$ , since the cylinder radius is fixed.
$\dot{V} = 4 π \dot{h}$ , giving $\dot{h} = 3 / (4 π)$ .
$3 / (4 π)$ m/min
A spherical balloon radius is 6 cm when its volume grows at $72 π$ cm $3$ /s. Find its radius rate.
2 marks
$\dot{V} = 4 π r^{2} \dot{r}$ .
$72 π = 144 π \dot{r}$ , so $\dot{r} = 1 / 2$ .
$1 / 2$ cm/s

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist Mathematics topic 6.1–6.2, with the approved school-level rates, differential-equation and motion applications.

Official curriculum source ↗

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