The perimeter of a playing card is 30 centimeters. The area is 54 square centimeters. What are the dimensions of the playing card? _____ centimeters by _____ centimeters

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The perimeter of a playing card is 30 centimeters.  The area is 54 square centimeters. What are the dimensions of the playing card?  _____ centimeters by _____ centimeters

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Define Card Dimensions: Let the length of the playing card be l centimeters and the width be w centimeters. The perimeter of a rectangle is given by the formula P = 2l + 2w.Perimeter Equation: Given the perimeter of the playing card is 30 centimeters, we can write the equation: 30 = 2l + 2wSimplify Perimeter: Simplify the perimeter equation by dividing all terms by 2 to make it easier to work with: 15 = l + wArea Calculation: The area of a rectangle is given by the formula A = lw. Given the area of the playing card is 54 square centimeters, we can write the equation: 54 = lwSystem of Equations: We now have a system of two equations with two variables: 15 = l + w (Equation 1) 54 = lw (Equation 2) We can solve this system by expressing one variable in terms of the other using Equation 1 and then substituting into Equation 2.Express Width in Terms of Length: From Equation 1, express w in terms of l: w = 15 - lSubstitute Width into Area Equation: Substitute w = 15 - l into Equation 2: 54 = l(15 - l)Expand and Rearrange Equation: Expand the equation and rearrange it into a quadratic equation: 54 = 15l - l^2 0 = l^2 - 15l + 54Factor Quadratic Equation: Factor the quadratic equation: 0 = (l - 9)(l - 6)Solve for Length: Solve for l by setting each factor equal to zero: l - 9 = 0 or l - 6 = 0 l = 9 or l = 6Find Possible Solutions: Since l and w are interchangeable in a rectangle (it doesn't matter which one is length and which one is width), we can have two solutions: If l = 9, then w = 15 - 9 = 6. If l = 6, then w = 15 - 6 = 9.Check Validity of Solutions: Check the solutions by verifying that they satisfy both the perimeter and area equations: For l = 9 and w = 6: Perimeter check: 2l + 2w = 2(9) + 2(6) = 18 + 12 = 30 (matches given perimeter) Area check: lw = 9 * 6 = 54 (matches given area)

The perimeter of a playing card is $30$ centimeters. The area is $54$ square centimeters. What are the dimensions of the playing card? $\newline$ $\_\_\_$ centimeters by $\_\_\_$ centimeters

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The perimeter of a playing card is $30$ centimeters. The area is $54$ square centimeters. What are the dimensions of the playing card? $\newline$ $\_\_\_$ centimeters by $\_\_\_$ centimeters