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Sagot :
Explanation
From the statement, we have a rectangular-shaped garden that:
• has a width ,w,,
,• has a length ,l = w + 2ft,,
,• an area ,A = 143 ft²,.
The area of a rectangle is given by:
[tex]A=w\cdot l.[/tex](1) Replacing the data from above, we have:
[tex]143ft^2=w\cdot(w+2ft).[/tex]Rewriting this equation, we get:
[tex]\begin{gathered} 143ft^2=w^2+2ft\cdot w, \\ w^2+2ft\cdot w-143ft^2, \\ w^2+2w-143=0. \end{gathered}[/tex]In the last equation, we have omitted the units.
(2) We know that the roots of a 2nd order polynomial equation:
[tex]a\cdot w^2+b\cdot w+c=0.[/tex]Are given by the formula:
[tex]w_{\pm}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.[/tex]We identify the coefficients:
• a = 1,
,• b = 2,
,• c = -143.
Replacing these coefficients in the formula above, we get:
[tex]\begin{gathered} w_+=\frac{-2+\sqrt{2^2-4\cdot2\cdot(-143)}}{2\cdot1}=11, \\ w_-=\frac{-2-\sqrt{2^2-4\cdot2\cdot(-143)}}{2\cdot1}=-13. \end{gathered}[/tex]Because w is the width, it can only take positive values, so we conclude that:
[tex]w=11.[/tex]Replacing this value in the equation for the length, we get:
[tex]l=11ft+2ft=13ft.[/tex]AnswerWidth = 11 ft and length = 13 ft
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