# Question #d590d

Oct 6, 2016

Seth bought 24 feet and Luke bought 27 feet of fencing

#### Explanation:

Let
$\textcolor{w h i t e}{\text{XXX}} {L}_{L}$ be the length of Luke's garden
$\textcolor{w h i t e}{\text{XXX}} {W}_{L}$ be the width of Luke's garden
$\textcolor{w h i t e}{\text{XXX}} {A}_{L}$ we the area of Luke's garden
$\textcolor{w h i t e}{\text{XXX}} {L}_{S}$ the the length of Seth's garden
$\textcolor{w h i t e}{\text{XXX}} {W}_{S}$ be the width of Seth's garden
$\textcolor{w h i t e}{\text{XXX}} {A}_{S}$ be the area of Seth's garden

Since we are told Seth's garden is square
$\textcolor{w h i t e}{\text{XXX}} {L}_{S} = {W}_{S}$
and
$\textcolor{w h i t e}{\text{XXX}} {A}_{S} = {\left({L}_{S}\right)}^{2}$
Further we are told that
$\textcolor{w h i t e}{\text{XXX}} {L}_{S} = {L}_{L} - 3$
So
$\textcolor{w h i t e}{\text{XXX}} {A}_{S} = {\left({L}_{L} - 3\right)}^{2} = {L}_{L}^{2} - 6 {L}_{L} + 9$

We are also told that
$\textcolor{w h i t e}{\text{XXX}} {W}_{L} = {L}_{L} / 2$
So
$\textcolor{w h i t e}{\text{XXX}} {A}_{L} = {L}_{L} \times {W}_{L} = \frac{{L}_{L}^{2}}{2}$

Since the areas are equal
$\textcolor{w h i t e}{\text{XXX}} {L}_{L}^{2} - 6 {L}_{L} + 9 = {L}_{L}^{2} / 2$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow 2 {L}_{L}^{2} - 12 {L}_{L} + 18 = {L}_{L}^{2}$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow {L}_{L}^{2} - 12 {L}_{L} + 18 = 0$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow \left({L}_{L} - 9\right) \left({L}_{L} - 3\right) = 0$
and
either ${L}_{L} = 9$ or ${L}_{L} = 3$
Since ${L}_{S} = {L}_{L} - 3$ and assuming Seth's garden is not $0$ feet long
${L}_{L} = 3$ must be extraneous
$\textcolor{w h i t e}{\text{XXX}} \Rightarrow {L}_{L} = 9$

Luke's garden
Since ${W}_{L} = {L}_{L} / 2$
the perimeter of Luke's garden must be
$\textcolor{w h i t e}{\text{XXX}} 2 \times \left(9 + \frac{9}{2}\right) = 27$ feet.

Seth's garden
Since ${L}_{S} = {L}_{L} - 3 \rightarrow {L}_{S} = 6$
and ${L}_{S} = {W}_{S}$
the perimeter of Seth's garden must be
$\textcolor{w h i t e}{\text{XXX}} 2 \times \left(6 + 6\right) = 24$ feet