A rectangular red sticker has an area of 18 square millimeters and a perimeter of 18 millimeters. What are the dimensions of the sticker? _____ millimeters by _____ millimeters

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A rectangular red sticker has an area of 18 square millimeters and a perimeter of 18 millimeters. What are the dimensions of the sticker?  _____ millimeters by _____ millimeters

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Define Variables: Let l be the length and w be the width of the rectangle. The area of a rectangle is given by the formula A = l * w.Area and Perimeter Formulas: The area of the sticker is given as 18 square millimeters. Substitute 18 for A in the formula A = l * w. 18 = l * wSubstitute Values: The perimeter of a rectangle is given by the formula P = 2l + 2w. The perimeter of the sticker is given as 18 millimeters. Substitute 18 for P in the formula P = 2l + 2w. 18 = 2l + 2wSimplify Perimeter Equation: Divide the perimeter equation by 2 to simplify. 18 / 2 = (2l + 2w) / 2 9 = l + wSolve System of Equations: We now have two equations with two variables: 18 = l * w (Area equation) 9 = l + w (Perimeter equation) 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 - lSubstitute into Area Equation: Substitute w = 9 - l into the area equation. 18 = l * (9 - l)Expand and Rearrange Equation: Expand the area equation and move all terms to one side to form a quadratic equation. 18 = 9l - l^2 0 = l^2 - 9l + 18Factor Quadratic Equation: Factor the quadratic equation. 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 = 6Final 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).

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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