Simple area and financial/application models | Year 9

01 / LEARN

The idea, step by step

Define the quantity first

If a rectangle is $x$ cm wide and $x + 5$ cm long, its area is $A = x (x + 5)$ . The product of two varying lengths creates a quadratic. Before solving, record what $x$ measures and which values make sense: lengths must be positive.

If its area is $84 {cm}^{2}$ , solve $x^{2} + 5 x - 84 = 0$ . The factors are $(x + 12) (x - 7)$ , so the algebra gives $x = - 12$ or $7$ . Only $7$ can be the width. The dimensions are $7$ cm by $12$ cm.

A mathematical model has limits

Financial models may also multiply changing quantities. If $n$ items each sell for $(12 - n)$ dollars, revenue is $R = n (12 - n)$ . Here $n$ is a whole-number count and the price should be non-negative. Different counts can produce the same revenue.

After solving, substitute each feasible result into the original situation. A model describes the stated assumptions, not a guaranteed real market. Do not interpret a negative count, impossible price or unsuitable measurement as a practical answer.

A common trap

Every algebraic root is a valid practical answer.

Instead: Check the meaning and units of the variable, then reject roots outside the allowed values.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example

A rectangle has width $x$ cm and length $x + 5$ cm. Its area is $84 {cm}^{2}$ . Find its dimensions.

$x (x + 5) = 84$ , so $x^{2} + 5 x - 84 = 0$ .
$(x + 12) (x - 7) = 0$ gives $x = - 12$ or $7$ .
Reject the negative width; check $7 \times 12 = 84$ .

7 cm

12 cm

Example 2

Worked example

Revenue is $R = n (12 - n)$ dollars for a whole number $n$ of items. Find counts giving $R = 32$ .

$12 n - n^{2} = 32$ , so $n^{2} - 12 n + 32 = 0$ .
$(n - 4) (n - 8) = 0$ .
Check $4 (8) = 32$ and $8 (4) = 32$ ; both counts give positive prices.

n = 4

8

Check your understanding

A square has area $49 {cm}^{2}$ . Algebra gives $x = \pm 7$ . What is its side length?

Show answer

$7 cm$

A physical side length is positive.

03 / WORKSHEET

Your turn

6 questions · 16 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A rectangle has width $x$ m and length $x + 3$ m. Write its area as an expanded expression.
2 marks
A rectangle is $x$ m wide and $x + 3$ m long, with area $40 m^{2}$ . Find its dimensions.
3 marks
Two consecutive positive integers have product $56$ . Find them.
3 marks
A model gives revenue $R = n (10 - n)$ . Find the revenue when $n = 3$ .
2 marks
For $R = n (10 - n)$ , find whole-number counts giving revenue $24$ .
3 marks
A cost model is $C = n^{2} + 3 n + 10$ dollars. Find the non-negative whole number of items when $C = 50$ .
3 marks

Read the worked solutions

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

A rectangle has width $x$ m and length $x + 3$ m. Write its area as an expanded expression.
2 marks
$x (x + 3) = x^{2} + 3 x$ .
$A = (x^{2} + 3 x) m^{2}$
A rectangle is $x$ m wide and $x + 3$ m long, with area $40 m^{2}$ . Find its dimensions.
3 marks
$x^{2} + 3 x - 40 = (x + 8) (x - 5) = 0$ .
Reject $x = - 8$ ; use $x = 5$ .
$5 m$ by $8 m$
Two consecutive positive integers have product $56$ . Find them.
3 marks
$n (n + 1) = 56$ , so $(n + 8) (n - 7) = 0$ .
Only $n = 7$ is positive.
$7$ and $8$
A model gives revenue $R = n (10 - n)$ . Find the revenue when $n = 3$ .
2 marks
$R = 3 (10 - 3) = 3 \times 7 = 21$ .
$21$
For $R = n (10 - n)$ , find whole-number counts giving revenue $24$ .
3 marks
$n^{2} - 10 n + 24 = (n - 4) (n - 6) = 0$ .
Both counts give a positive price.
$n = 4$ or $6$
A cost model is $C = n^{2} + 3 n + 10$ dollars. Find the non-negative whole number of items when $C = 50$ .
3 marks
$n^{2} + 3 n - 40 = (n + 8) (n - 5) = 0$ .
Reject $n = - 8$ ; check $25 + 15 + 10 = 50$ .
$n = 5$

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

AC9M9A02, AC9M9A04–A06: monic quadratic expressions, integer-root equations, graphs and models.

Official curriculum source ↗

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