A rectangular patio has an area of 91 fl. The length is 6 feetlonger than the width. Find the dimensions of the patio areaSolve by completing the square, Find the width and the length interms of w. Write an equation for the total area Find b/2 Find the dimensions.

Accepted Solution

Answer:Total area equation = tex]w(w+6)=91[/tex]b/2 = 3Dimensions of the patio: width = 7 feet, length = 13 feetStep-by-step explanation:The area of a rectangle is given the formula:[tex]A=wl[/tex]where[tex]w[/tex] is the width [tex]l[/tex] is the length We know from our problem that the area of the patio is 91 square feet, so [tex]A=91[/tex]. We also know that the length is 6 feet longer then the width, so [tex]l=w+6[/tex].Replacing values in our area equation[tex]A=wl[/tex][tex]91=w(w+6)[/tex][tex]w(w+6)=91[/tex]Expanding the left side:[tex]w*w+6w=91[/tex][tex]w^2+6w=91[/tex]Remember that to complete the square we need to add half the coefficient of the linear term squared. The lineal term is [tex]w[/tex], so its coefficient is 6. Now, half its coefficient or [tex]\frac{b}{2} =\frac{6}{2} =3[/tex]. Finally, [tex]3^2=9[/tex]. To complete the square we need to add 9 to both sides of the equation:[tex]w^2+6w+9=91+9[/tex][tex]w^2+6w+9=100[/tex]Notice that the left side is a perfect square trinomial (both [tex]w^2[/tex] and 9 are perfect squares), so we can express it as:[tex](w+3)^2=100[/tex]Now that we completed the square, we can solve our equation - Take square root to both sides [tex]\sqrt{(w+3)^2} =\pm\sqrt{100}[/tex][tex]w+3=\pm10[/tex]- Subtract 3 from both results [tex]w=10-3,w=-10-3[/tex][tex]w=7,w=13[/tex]Since length cannot be negative, [tex]w=7[/tex] is the solution of our equation. We now know that the width of our rectangular patio is 7 feet, so we can find its length: [tex]l=w+6[/tex][tex]l=7+6[/tex][tex]l=13[/tex]We can conclude that half the coefficient of the width is [tex]\frac{b}{2}=3[/tex], the width of the patio is 7 feet, and its length is 13 feet.