A rectangular sticker has an area of $28\,\text{square centimeters}$ and a perimeter of $22\,\text{centimeters}$ . What are the dimensions of the sticker? $\newline$ $\_\_\_\_$ centimeters by $\_\_\_\_$ centimeters

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Area and perimeter: word problems

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Q. A rectangular sticker has an area of

28\,\text{square centimeters}

and a perimeter of

22\,\text{centimeters}

. What are the dimensions of the sticker?

\newline

\_\_\_\_

centimeters by

\_\_\_\_

centimeters

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 $28$ square centimeters. $\newline$ Substitute $28$ for $A$ in the formula $A = l \times w$ . $\newline$ $28 = l \times w$
Substitute Values: The perimeter of a rectangle is given by the formula $P = 2l + 2w$ . $\newline$ The perimeter of the sticker is given as $22$ centimeters. $\newline$ Substitute $22$ for $P$ in the formula $P = 2l + 2w$ . $\newline$ $22 = 2l + 2w$
Simplify Perimeter: Divide the perimeter equation by $2$ to simplify. $\newline$ $\frac{22}{2} = \frac{2l + 2w}{2}$ $\newline$ $11 = l + w$
Solve System of Equations: We now have a system of two equations with two variables: $\newline$ $28 = l \times w$ (Area equation) $\newline$ $11 = l + w$ (Perimeter equation) $\newline$ We can solve this system by expressing one variable in terms of the other using the perimeter equation and then substituting into the area equation.
Express Width in Terms of Length: Express $w$ in terms of $l$ using the perimeter equation. $w = 11 - l$
Substitute Width into Area Equation: Substitute $w = 11 - l$ into the area equation $28 = l \times w$ . $\newline$ $28 = l \times (11 - l)$
Expand and Rearrange Equation: Expand the equation and move all terms to one side to form a quadratic equation. $\newline$ $28 = 11l - l^2$ $\newline$ $0 = l^2 - 11l + 28$
Factor Quadratic Equation: Factor the quadratic equation. $\newline$ $0 = (l - 4)(l - 7)$
Solve for Length: Solve for $l$ by setting each factor equal to zero. $l - 4 = 0$ or $l - 7 = 0$ $l = 4$ or $l = 7$
Find Possible Dimensions: If $l = 4$ , then $w = 11 - l = 11 - 4 = 7$ . If $l = 7$ , then $w = 11 - l = 11 - 7 = 4$ . So the dimensions of the sticker can be either $4 \, \text{cm} \times 7 \, \text{cm}$ or $7 \, \text{cm} \times 4 \, \text{cm}$ .