Simple area and financial/application models | Year 9

01 / LEARN

Understand the idea.

Define the quantity first

Begin an application by defining the variable and its feasible values. A rectangle's area is width times length, so lengths $x$ and $x + 5$ produce $A = x (x + 5)$ . A physical width must be positive.

Set the model equal to the required area, rearrange to zero, and factorise. The resulting algebraic roots are candidates. Substitute the feasible one into both dimensions and verify that their product has the stated area and units.

A mathematical model has limits

Revenue is count multiplied by price per item. If the price depends on the count, as in $12 - n$ dollars, the product $n (12 - n)$ is quadratic. A requested revenue can therefore correspond to two different counts.

Check every candidate against the context: counts are non-negative whole numbers and the modelled price should not be negative. A supplied revenue rule represents its assumptions, so satisfying the algebra alone does not guarantee a feasible sale.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example

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

Worked solution

1
Form the area equation.
$x (x + 5) = 84, x > 0$
Rectangle area is the product of its positive side lengths, measured here in square centimetres.
2
Expand and rearrange to zero.
$x^{2} + 5 x - 84 = 0$
Subtracting 84 prepares the quadratic for the zero-product principle.
3
Factorise and solve both cases.
$\begin{aligned} (x + 12) (x - 7) & = 0 \\ x & = - 12 or x = 7 \end{aligned}$
The constants 12 and -7 multiply to -84 and add to 5.
4
Reject the negative width and find the length.
$x = 7 cm, x + 5 = 12 cm$
A physical side length cannot be negative.
5
Check the dimensions.
$7 \times 12 = 84 {cm}^{2}$
The accepted dimensions reproduce the original area.

Answer

$7 cm$ by $12 cm$

EXAMPLE 02

Worked example

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

Worked solution

1
Set the target revenue.
$n (12 - n) = 32$
The product of count and modelled price must equal 32 dollars.
2
Write a quadratic equal to zero.
$\begin{aligned} 12 n - n^{2} & = 32 \\ n^{2} - 12 n + 32 & = 0 \end{aligned}$
Moving all terms and changing signs gives a monic quadratic.
3
Factorise and solve.
$\begin{aligned} (n - 4) (n - 8) & = 0 \\ n & = 4 or n = 8 \end{aligned}$
The two constants multiply to 32 and sum to -12.
4
Check both counts and prices.
$\begin{aligned} n = 4 : & 12 - n = 8, 4 (8) = 32 \\ n = 8 : & 12 - n = 4, 8 (4) = 32 \end{aligned}$
Both counts are whole and non-negative, and each gives a positive price and the required revenue.

Answer

$n = 4$ or $8$

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$7 cm$

Although $x^{2} = 49$ has roots $x = \pm 7$ , a square's side length must be positive. Its side is therefore $7 cm$ , and $7^{2} = 49 {cm}^{2}$ .

Keep practising

6 worksheet questions · 16 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Define the quantity first

A mathematical model has limits

See the method in action.

Worked example

Form the area equation.

Expand and rearrange to zero.

Factorise and solve both cases.

Reject the negative width and find the length.

Check the dimensions.

Worked example

Set the target revenue.

Write a quadratic equal to zero.

Factorise and solve.

Check both counts and prices.

Try it yourself.

Check your understanding

Keep practising