A rectangular red sticker has an area of $18 \, \text{square millimeters}$ and a perimeter of $18 \, \text{millimeters}$ . What are the dimensions of the sticker? $\newline$ $\_\_\_\_\_$ millimeters by $\_\_\_\_\_$ millimeters

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Area and perimeter: word problems

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Q. A rectangular red sticker has an area of

18 \, \text{square millimeters}

and a perimeter of

18 \, \text{millimeters}

. What are the dimensions of the sticker?

\newline

\_\_\_\_\_

millimeters by

\_\_\_\_\_

millimeters

Define Variables: Let $l$ be the length and $w$ be the width of the rectangle. $\newline$ The area of a rectangle is given by the formula $A = l \times w$ .
Area and Perimeter Formulas: The area of the sticker is given as $18$ square millimeters. $\newline$ Substitute $18$ for $A$ in the formula $A = l \times w$ . $\newline$ $18 = l \times w$
Substitute Values: The perimeter of a rectangle is given by the formula $P = 2l + 2w$ . $\newline$ The perimeter of the sticker is given as $18$ millimeters. $\newline$ Substitute $18$ for $P$ in the formula $P = 2l + 2w$ . $\newline$ $18 = 2l + 2w$
Simplify Perimeter Equation: Divide the perimeter equation by $2$ to simplify. $\newline$ $\frac{18}{2} = \frac{2l + 2w}{2}$ $\newline$ $9 = l + w$
Solve System of Equations: We now have two equations with two variables: $\newline$ $18 = l \times w$ (Area equation) $\newline$ $9 = l + w$ (Perimeter equation) $\newline$ We can solve this system of equations to find the values of $l$ and $w$ .
Express Width in Terms of Length: From the perimeter equation, express $w$ in terms of $l$ . $w = 9 - l$
Substitute into Area Equation: Substitute $w = 9 - l$ into the area equation. $18 = l \times (9 - l)$
Expand and Rearrange Equation: Expand the area equation and move all terms to one side to form a quadratic equation. $\newline$ $18 = 9l - l^2$ $\newline$ $0 = l^2 - 9l + 18$
Factor Quadratic Equation: Factor the quadratic equation. $\newline$ $0 = (l - 3)(l - 6)$
Solve for Length: Solve for $l$ by setting each factor equal to zero. $l - 3 = 0$ or $l - 6 = 0$ $l = 3$ or $l = 6$
Final Dimensions: If $l = 3$ , then $w = 9 - l = 9 - 3 = 6$ . If $l = 6$ , then $w = 9 - l = 9 - 6 = 3$ . So the dimensions of the sticker can be $3$ millimeters by $6$ millimeters or $6$ millimeters by $3$ millimeters (since a rectangle's length and width are interchangeable).