ScratchpadThe chicken coup at the petting zoo is 10 by 14 feet. The farm would like to double the current area by adding thesame amount, x, to the length and width. What are the dimensions of the new enclosure? Round to the nearesthundredth of a foot.

Accepted Solution

Answer:New width of coup [tex]=14.85[/tex] feetNew length of coup [tex]=18.85[/tex] feetStep-by-step explanation:Width of chicken coup =[tex]10[/tex] feetLength of chicken coup=[tex]14[/tex] feetArea of the coup = [tex]length\times width = 10\times14=140\ ft^2[/tex]New area is double the current area which is = [tex]2\times140=280\ ft^2[/tex][tex]x[/tex] is added to both length and width.New width of coup = [tex]10+x[/tex] feetNew length of coup = [tex]14+x[/tex] feetNew area of coup can be written as = [tex]length\times width =(10+x)\times(14+x)[/tex]So, we have[tex](10+x)(14+x)=280[/tex]Using FOIL method to multiply.[tex](10\times14)+10x+14x+x^2=280\\140+24x+x^2=280[/tex]Subtracting 280 both sides.[tex]140+24x+x^2-280=280-280[/tex]Rearranging terms, we have the quadratic equation to solve.[tex]x^2+24x-140=0[/tex]Using quadratic formula to find [tex]x[/tex][tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]Plugging in values.[tex]x=\frac{-24\pm\sqrt{24^2-(4(1)(-140))}}{2(1)}[/tex][tex]x = \frac{ -24\pm\sqrt{576-(-560)}}{2}\\[/tex][tex]x = \frac{ -24\pm\sqrt{1136}}{2}[/tex][tex]x = \frac{-24\pm 4\sqrt{71}}{2}[/tex][tex]x = \frac{-24}{2} \pm \frac{4\sqrt{71}}{2}[/tex][tex]\therefore x=4.8523\approx 4.85\ and\ x=-28.8523\approx -28.85[/tex]Since [tex]x[/tex] is the length, so it cannot be negative. So [tex]x=4.85[/tex]New width of coup = [tex]10+x=10+4.85=14.85[/tex] feetNew length of coup = [tex]14+x=14+4.85=18.85[/tex] feet