A chalkboard has a perimeter of 16 feet. Its area is 15 square feet. What are the dimensions of the board? _____ feet by _____ feet

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A chalkboard has a perimeter of 16 feet.  Its area is 15 square feet. What are the dimensions of the board?  _____ feet by _____ feet

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Define Variables: Let l be the length and w be the width of the chalkboard. The perimeter of a rectangle is given by the formula P = 2l + 2w.Perimeter Equation: Given the perimeter of the chalkboard is 16 feet, we can write the equation 16 = 2l + 2w.Simplify Perimeter: Simplify the perimeter equation by dividing all terms by 2 to get 8 = l + w.Area Equation: The area of a rectangle is given by the formula A = lw. Given the area of the chalkboard is 15 square feet, we can write the equation 15 = lw.Solve System: We now have a system of two equations with two variables: 1. 8 = l + w 2. 15 = lw  We can solve this system by expressing one variable in terms of the other using the first equation and then substituting into the second equation.Express w in terms of l: From the first equation, express w in terms of l: w = 8 - l.Substitute into Area Equation: Substitute w = 8 - l into the second equation 15 = lw to get 15 = l(8 - l).Expand Equation: Expand the equation 15 = l(8 - l) to get 15 = 8l - l^2.Form Quadratic Equation: Rearrange the equation to form a quadratic equation: l^2 - 8l + 15 = 0.Factor Quadratic Equation: Factor the quadratic equation to find the values of l: (l - 3)(l - 5) = 0.Solve for l: Set each factor equal to zero and solve for l: l - 3 = 0 or l - 5 = 0. This gives us l = 3 or l = 5.Find Dimensions: If l = 3, then w = 8 - l = 8 - 3 = 5. If l = 5, then w = 8 - l = 8 - 5 = 3. So the dimensions of the chalkboard can be either 3 feet by 5 feet or 5 feet by 3 feet.

A chalkboard has a perimeter of $16\,\text{feet}$ . Its area is $15\,\text{square feet}$ . What are the dimensions of the board? $\newline$ $\_\_\_\_$ feet by $\_\_\_\_$ feet

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A chalkboard has a perimeter of 16 feet16\,\text{feet}16feet. Its area is 15 square feet15\,\text{square feet}15square feet. What are the dimensions of the board?\newline____\_\_\_\_____ feet by ____\_\_\_\_____ feet

A chalkboard has a perimeter of $16\,\text{feet}$ . Its area is $15\,\text{square feet}$ . What are the dimensions of the board? $\newline$ $\_\_\_\_$ feet by $\_\_\_\_$ feet