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A poster has a perimeter of 1616 feet and an area of 1515 square feet. What are the dimensions of the poster?\newline____\_\_\_\_ feet by ____\_\_\_\_ feet

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Q. A poster has a perimeter of 1616 feet and an area of 1515 square feet. What are the dimensions of the poster?\newline____\_\_\_\_ feet by ____\_\_\_\_ feet
  1. Define Variables: Let ll be the length and ww be the width of the poster.\newlineThe perimeter of a rectangle is given by the formula P=2l+2wP = 2l + 2w.
  2. Perimeter Equation: Given the perimeter of the poster is 1616 feet, we can write the equation 16=2l+2w16 = 2l + 2w.
  3. Simplify Perimeter: Simplify the equation by dividing all terms by 22 to get 8=l+w8 = l + w.
  4. Area Equation: The area of a rectangle is given by the formula A=lwA = lw.\newlineGiven the area of the poster is 1515 square feet, we can write the equation 15=lw15 = lw.
  5. Solve System of Equations: We now have a system of two equations with two variables:\newline11. 8=l+w8 = l + w\newline22. 15=lw15 = lw\newlineWe can solve this system of equations to find the values of ll and ww.
  6. Express ww in terms of ll: From the first equation, we can express ww in terms of ll: w=8−lw = 8 - l.
  7. Substitute into Area Equation: Substitute w=8−lw = 8 - l into the second equation: 15=l(8−l)15 = l(8 - l).
  8. Expand and Rearrange: Expand the equation: 15=8l−l215 = 8l - l^2.
  9. Form Quadratic Equation: Rearrange the equation to form a quadratic equation: l2−8l+15=0l^2 - 8l + 15 = 0.
  10. Factor Quadratic Equation: Factor the quadratic equation: (l−5)(l−3)=0(l - 5)(l - 3) = 0.
  11. Solve for l: Solve for l: l=5l = 5 or l=3l = 3.
  12. Find Dimensions: If l=5l = 5, then w=8−l=8−5=3w = 8 - l = 8 - 5 = 3. If l=3l = 3, then w=8−l=8−3=5w = 8 - l = 8 - 3 = 5.
  13. Find Dimensions: If l=5l = 5, then w=8−l=8−5=3w = 8 - l = 8 - 5 = 3. If l=3l = 3, then w=8−l=8−3=5w = 8 - l = 8 - 3 = 5.We have two possible dimensions for the poster: 55 feet by 33 feet or 33 feet by 55 feet. Both sets of dimensions satisfy the given perimeter and area.

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