Answer
The rectangular sign is $4$ feet in width and $10$ feet in length.
Work Step by Step
To solve this problem, we translate it into an algebraic equation that we can solve. We do this by going bit by bit through the problem and translating each part and then putting all the parts together into an equation.
We know that the formula of the perimeter of a rectangle is given by:
$$P = 2l + 2w$$
where $l$ is the length and $w$ is the width.
If $w$ is the width, then the length is:
"$2$ feet less than three times its width" means the length is "$3w - 2$".
If the perimeter is $28$ feet, then we can plug in these values into the equation for perimeter:
$$28 = 2(3w - 2) + 2w$$
Use distributive property:
$$28 = 6w - 4 + 2w$$
Combine the variables:
$$8w - 4 = 28$$
Add $4$ to each side to isolate the constants to one side of the equation:
$$8w = 32$$
Divide each side of the equation by $8$ to solve for $w$:
$$w = 4$$
If we now know that the width of the rectangular sign is $4$ feet, then we can find the length by plugging $4$ into the expression for length, which was given as $3w - 2$.
$$l = 3(4) - 2$$
Multiply first, according to order of operations:
$$l = 12 - 2$$
Subtract to solve for $l$:
$$l = 10$$
The rectangular sign is $4$ feet in width and $10$ feet in length.
