Question #d590d

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Question #d590d

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Answer 1 · 2016-10-06T13:00:39

Seth bought 24 feet and Luke bought 27 feet of fencing

Explanation:

Let
$XXX L_{L}$ $color(white)("XXX")L_L$ be the length of Luke's garden
$XXX W_{L}$ $color(white)("XXX")W_L$ be the width of Luke's garden
$XXX A_{L}$ $color(white)("XXX")A_L$ we the area of Luke's garden
$XXX L_{S}$ $color(white)("XXX")L_S$ the the length of Seth's garden
$XXX W_{S}$ $color(white)("XXX")W_S$ be the width of Seth's garden
$XXX A_{S}$ $color(white)("XXX")A_S$ be the area of Seth's garden

Since we are told Seth's garden is square
$XXX L_{S} = W_{S}$ $color(white)("XXX")L_S=W_S$
and
$XXX A_{S} = {(L_{S})}_{S}^{2}$ $color(white)("XXX")A_S=(L_S)^2$
Further we are told that
$XXX L_{S} = L_{L} - 3$ $color(white)("XXX")L_S=L_L-3$
So
$XXX A_{S} = {(L_{L} - 3)}_{L}^{2} = L_{L}^{2} - 6 L_{L} + 9$ $color(white)("XXX")A_S=(L_L-3)^2=L_L^2-6L_L+9$

We are also told that
$XXX W_{L} = \frac{L_{L}}{2}$ $color(white)("XXX")W_L=L_L/2$
So
$XXX A_{L} = L_{L} \times W_{L} = \frac{L_{L}^{2}}{2}$ $color(white)("XXX")A_L=L_L xx W_L = (L_L^2)/2$

Since the areas are equal
$XXX L_{L}^{2} - 6 L_{L} + 9 = \frac{L_{L}^{2}}{2}$ $color(white)("XXX")L_L^2-6L_L+9=L_L^2/2$

$XXX \to 2 L_{L}^{2} - 12 L_{L} + 18 = L_{L}^{2}$ $color(white)("XXX")rarr2L_L^2-12L_L+18=L_L^2$

$XXX \to L_{L}^{2} - 12 L_{L} + 18 = 0$ $color(white)("XXX")rarr L_L^2-12L_L+18=0$

$XXX \to (L_{L} - 9) (L_{L} - 3) = 0$ $color(white)("XXX")rarr (L_L-9)(L_L-3)=0$
and
either $L_{L} = 9$ $L_L=9$ or $L_{L} = 3$ $L_L=3$
Since $L_{S} = L_{L} - 3$ $L_S=L_L-3$ and assuming Seth's garden is not $0$ $0$ feet long
$L_{L} = 3$ $L_L=3$ must be extraneous
$XXX \Rightarrow L_{L} = 9$ $color(white)("XXX")rArr L_L=9$

Luke's garden
Since $W_{L} = \frac{L_{L}}{2}$ $W_L=L_L/2$
the perimeter of Luke's garden must be
$XXX 2 \times (9 + \frac{9}{2}) = 27$ $color(white)("XXX")2xx(9+9/2)=27$ feet.

Seth's garden
Since $L_{S} = L_{L} - 3 \to L_{S} = 6$ $L_S=L_L-3 rarr L_S=6$
and $L_{S} = W_{S}$ $L_S=W_S$
the perimeter of Seth's garden must be
$XXX 2 \times (6 + 6) = 24$ $color(white)("XXX")2xx(6+6)=24$ feet

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Question #d590d

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