Question 315122
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If the inner rectangle (the playground part) is 10 by 14 and we let *[tex \Large x] represent the width of the grass border, then the overall size of the rectangle is *[tex \Large 10\ +\ 2x] by *[tex \Large 14\ +\ 2x] because the walkway is on both sides and on the top and bottom, so two widths of the walkway in each direction.


The area of just the playground is *[tex \Large 10\ \times\ 14\ =\ 140\text{ ft^2}]


The area of just the grass border is given as *[tex \Large 145\text{ ft^2}]


So the total area must be *[tex \Large 140\ +\ 145\ =\ 285\text{ ft^2}]


But since the overall dimensions are *[tex \Large 10\ +\ 2x] by *[tex \Large 14\ +\ 2x], the total area must also be represented by:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left(10\ +\ 2x\right)\left(14\ +\ 2x\right)\ =\ 4x^2\ +\ 48x\ +\ 140]


Which we have determined is equal to 285, so:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 4x^2\ +\ 48x\ +\ 140\ =\ 285]


Hence


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 4x^2\ +\ 48x\ -\ 145\ =\ 0]


Fortunately for those of us who enjoy nice neat rational number answers, this little quadratic factors rather neatly:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left(2x\ -\ 5\right)\left(2x\ +\ 29\right)\ =\ 0]


Which means that


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ 2.5\text{ ft}]


or


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ -14.5\text{ ft}]


The idea of a grass border that is -14.5 feet wide is simply ludicrous, so discard that root.  The correct answer is 2.5 feet.


<b><i>Check</i></b>


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 10\ +\ 2(2.5)\ =\ 15]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 14\ +\ 2(2.5)\ =\ 19]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 15\ \times\ 19\ =\ 285]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 285\ -\ 140\ =\ 145]


<b><i>Answer checks</i></b>



John
*[tex \LARGE e^{i\pi} + 1 = 0]
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