CH-05-001 · Quadratic modelling: rectangular border

Diagram not accurately drawn

A rectangular lawn is 16 m long and 9 m wide. A path of uniform width x metres surrounds it. The area of the path is 264 m². Work out the width of the path and the perimeter of its outside edge. You must use algebra.

Solving Quadratics by Factorising (Rectangular Border) Booklet p. 2, p. 6 & p. 12

Method

1) Form border area equation (L + 2x)(W + 2x) − LW = Area

2) Expand using 2×2 grid and rearrange = 0

3) Factorise/solve for positive x and find outside perimeter

Worked Example (CH-05-001)

Let x be the path width. Outer dimensions are

16+2x

and

9+2x

×	$16$	$2x$
$9$	$144$	$18x$
$2x$	$32x$	$4x^2$

Outer area − lawn area = path area:

4x^2+50x-264=0

Take out the HCF first:

4x^2 + 50x - 264=2(2x^2 + 25x - 132)

Multiply

a\times c=2\times(-132)=-264

. Choose the signed factor pair

-8,\ 33

: product

-264

, sum

25

×	$x$	$-4$
$2x$	$2x^2$	$-8x$
$33$	$33x$	$-132$

Read the factors outside the grid:

2(2x + 33)(x - 4)

2x + 33=0\quad\Rightarrow\quad x=-\frac{33}{2}

x - 4=0\quad\Rightarrow\quad x=4

10)

Reject the negative root: a width is positive, so

x=4

11)

Outside perimeter:

2[(16+2(4))+(9+2(4))]=82

Ans = Width 4 m; outside perimeter 82 m.