Question

find lengths using area and perimeter mastery check a rectangle has a perimeter of 18 yards and an area of 18 square yards. what are the length and width? length: yards width: yards

Explanation:

Step1: Set up equations

Let the length be $l$ and the width be $w$. The perimeter formula of a rectangle is $P = 2(l + w)$, so $2(l + w)=18$, which simplifies to $l + w=9$ (dividing both sides by 2), and we can express $l=9 - w$. The area formula is $A=lw$, so $lw = 18$.

Step2: Substitute and solve

Substitute $l = 9 - w$ into $lw=18$, we get $(9 - w)w=18$. Expand it to $9w-w^{2}=18$, or $w^{2}-9w + 18=0$. Factor the quadratic equation: $(w - 3)(w - 6)=0$.

Step3: Find the values of $w$ and $l$

Set each factor equal to zero: If $w - 3=0$, then $w = 3$; if $w - 6=0$, then $w = 6$. When $w = 3$, $l=9 - 3=6$; when $w = 6$, $l=9 - 6=3$.

Answer:

length: 6 yards
width: 3 yards
(or length: 3 yards, width: 6 yards)