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A rectangular bathroom mirror has an area of 2727 square feet and a perimeter of 2424 feet. What are the dimensions of the mirror?\newline___\_\_\_ feet by ___\_\_\_ feet

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Q. A rectangular bathroom mirror has an area of 2727 square feet and a perimeter of 2424 feet. What are the dimensions of the mirror?\newline___\_\_\_ feet by ___\_\_\_ feet
  1. Define Variables: Let's denote the length of the mirror as LL feet and the width as WW feet. We know that the area (A)(A) of a rectangle is given by A=L×WA = L \times W and the perimeter (P)(P) is given by P=2L+2WP = 2L + 2W. We are given that A=27A = 27 square feet and P=24P = 24 feet. We need to set up two equations based on these formulas and solve for LL and WW.
  2. Area Equation: First, let's write down the area equation with the given value:\newlineA=L×WA = L \times W\newline27=L×W27 = L \times W
  3. Perimeter Equation: Now, let's write down the perimeter equation with the given value:\newlineP=2L+2WP = 2L + 2W\newline24=2L+2W24 = 2L + 2W
  4. Simplify Perimeter: We can simplify the perimeter equation by dividing all terms by 22 to make it easier to solve:\newline24÷2=L+W24 \div 2 = L + W\newline12=L+W12 = L + W
  5. System of Equations: Now we have a system of two equations:\newline11) 27=L×W27 = L \times W\newline22) 12=L+W12 = L + W\newlineWe can solve this system by expressing one variable in terms of the other using the second equation and then substituting it into the first equation.
  6. Express WW in Terms of LL: Let's express WW in terms of LL using the second equation:\newlineW=12−LW = 12 - L\newlineNow we can substitute this expression for WW into the first equation.
  7. Substitute WW in Area Equation: Substituting WW in the area equation, we get:\newline27=L×(12−L)27 = L \times (12 - L)\newlineThis is a quadratic equation: L2−12L+27=0L^2 - 12L + 27 = 0\newlineWe need to solve this quadratic equation for LL.
  8. Solve Quadratic Equation: To solve the quadratic equation, we can factor it if possible:\newline(L−3)(L−9)=0(L - 3)(L - 9) = 0\newlineThis gives us two possible solutions for LL: L=3L = 3 or L=9L = 9.
  9. Factor Quadratic Equation: If L=3L = 3, then using the equation W=12−LW = 12 - L, we find that W=12−3=9W = 12 - 3 = 9. If L=9L = 9, then W=12−9=3W = 12 - 9 = 3. So the dimensions of the mirror can be 33 feet by 99 feet or 99 feet by 33 feet, which are essentially the same since it's a rectangle and the sides can be interchanged.

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