A garden area is 30 ft long and 20 ft wide. A path of uniform width is set inside the edge. If the remaining garden is 400ft^2, what is the width of the path?
Accepted Solution
A:
Answer:2.19 ft ( approx )Step-by-step explanation:Let x be the width ( in ft ) of the path,Given,The dimension of the garden area,Length = 30 ft, width = 20 ft,So, the dimension of the remaining garden ( garden excluded path ),Length = (30 - 2x) ft, width = (20-2x) ftThus, the area of the remaining garden,A=(30 - 2x)(20 - 2x)According to the question,A = 400 ft²,[tex](30 - 2x)(20 - 2x)=400[/tex][tex]600-60x -40x + 4x^2= 400[/tex][tex]4x^2-100x+600-400=0[/tex][tex]4x^2-100x+200=0[/tex][tex]x^2-25x+50=0[/tex]By the quadratic formula,[tex]x=\frac{-(-25)\pm \sqrt{(-25)^2-4\times 1\times 50}}{2}[/tex][tex]=\frac{25\pm \sqrt{625-200}}{2}[/tex][tex]=\frac{25\pm \sqrt{425}}{2}[/tex][tex]\implies x = \frac{25+ \sqrt{425}}{2}\text{ or }x=\frac{25- \sqrt{425}}{2}[/tex]⇒ x ≈ 22.8 or x ≈ 2.19,∵ Width of the path can not exceed 30 ft or 20 ftHence, the width of the path is approximately 2.19 ft.