Question

find lengths using area and perimeter mastery check the perimeter of a rectangle is 36 feet, and the area is 32 square feet. what are the length and width? length: feet width: feet

Explanation:

Step1: Set up equations

Let the length be $l$ and the width be $w$. The perimeter formula is $P = 2(l + w)$, so $2(l + w)=36$, which simplifies to $l + w=18$, or $l = 18 - w$. The area formula is $A=lw$, so $lw = 32$.

Step2: Substitute and solve

Substitute $l = 18 - w$ into $lw = 32$, we get $(18 - w)w=32$. Expand to $18w-w^{2}=32$, or $w^{2}-18w + 32=0$.

Step3: Factor the quadratic equation

Factor $w^{2}-18w + 32=0$ as $(w - 2)(w - 16)=0$.

Step4: Find the values of width and length

Set each factor equal to zero: $w - 2=0$ gives $w = 2$, then $l=18 - 2=16$; $w - 16=0$ gives $w = 16$, then $l=18 - 16=2$.

Answer:

length: 16 feet, width: 2 feet (or length: 2 feet, width: 16 feet)