Question

the perimeter of a rectangle is 86 feet. find the length and width if the length is an integer and the width is 3 times the next consecutive integer.

Explanation:

Step1: Define variables

Let the length of the rectangle be $x$ (where $x$ is an integer). The next - consecutive integer is $x + 1$, and the width $w=3(x + 1)$.

Step2: Use the perimeter formula

The perimeter formula of a rectangle is $P = 2(l+w)$. Given $P = 86$, we substitute $l=x$ and $w = 3(x + 1)$ into the formula:
$86=2(x + 3(x + 1))$.

Step3: Simplify the equation

First, expand the expression inside the parentheses: $86=2(x+3x + 3)$. Then combine like - terms inside the parentheses: $86=2(4x + 3)$. Distribute the 2: $86 = 8x+6$.

Step4: Solve for $x$

Subtract 6 from both sides of the equation: $86−6=8x$, so $80 = 8x$. Divide both sides by 8: $x = 10$.

Step5: Find the length and width

The length $l=x = 10$ feet. The width $w=3(x + 1)=3\times(10 + 1)=33$ feet.

Answer:

Length is 10 feet and width is 33 feet.