A rectangular sticker has a perimeter of $26$ centimeters. Its area is $36$ square centimeters. What are the dimensions of the sticker? $\newline$ $\_\_\_$ centimeters by $\_\_\_$ centimeters

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Q. A rectangular sticker has a perimeter of

26

centimeters. Its area is

36

square 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 rectangular sticker. $\newline$ The perimeter of a rectangle is given by the formula $P = 2l + 2w$ .
Perimeter Equation: Given the perimeter of the sticker is $26$ centimeters, we can write the equation $26 = 2l + 2w$ .
Simplify Perimeter: Simplify the perimeter equation by dividing all terms by $2$ to get $13 = l + w$ .
Area Formula: The area of a rectangle is given by the formula $A = lw$ . Given the area of the sticker is $36$ square centimeters, we can write the equation $36 = lw$ .
System of Equations: We now have a system of two equations: $\newline$ $1$ . $13 = l + w$ $\newline$ $2$ . $36 = lw$ $\newline$ We will solve this system to find the values of $l$ and $w$ .
Express Width: From the first equation, we can express $w$ in terms of $l$ : $w = 13 - l$ .
Substitute Width: Substitute $w = 13 - l$ into the second equation $36 = lw$ to get $36 = l(13 - l)$ .
Distribute Equation: Distribute $l$ across the parentheses: $36 = 13l - l^2$ .
Quadratic Form: Rearrange the equation into standard quadratic form: $0 = l^2 - 13l + 36$ .
Factor Equation: Factor the quadratic equation: $(l - 4)(l - 9) = 0$ .
Solve for $l$ : Solve for $l$ by setting each factor equal to zero: $l - 4 = 0$ or $l - 9 = 0$ . This gives us two possible values for $l$ : $l = 4$ or $l = 9$ .
Calculate Dimensions: If $l = 4$ , then $w = 13 - l = 13 - 4 = 9$ . If $l = 9$ , then $w = 13 - l = 13 - 9 = 4$ . So the dimensions of the sticker can be either $4 \, \text{cm} \times 9 \, \text{cm}$ or $9 \, \text{cm} \times 4 \, \text{cm}$ .