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The perimeter of a rectangular living room is $34$ meters. The area is $72$ square meters. What are the dimensions of the living room? $\newline$ $\_\_\_$ meters by $\_\_\_$ meters

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Area and perimeter: word problems

Full solution

Q. The perimeter of a rectangular living room is

34

meters. The area is

72

square meters. What are the dimensions of the living room?

\newline

\_\_\_

meters by

\_\_\_

meters

Define Perimeter Equation: Let the length be $l$ meters and the width be $w$ meters. The perimeter of a rectangle is given by $P = 2l + 2w$ .
Simplify Perimeter Equation: Given the perimeter is $34$ meters, we set up the equation $2l + 2w = 34$ . Simplify this by dividing everything by $2$ : $l + w = 17$ .
Define Area Equation: The area of a rectangle is given by $A = lw$ . We know the area is $72$ square meters, so $lw = 72$ .
Solve Equations Simultaneously: We now have two equations: $l + w = 17$ and $lw = 72$ . Solve these simultaneously. Start by expressing $w$ from the first equation: $w = 17 - l$ .
Substitute and Simplify: Substitute $w = 17 - l$ into the area equation: $l(17 - l) = 72$ . This simplifies to $17l - l^2 = 72$ . Rearrange to form a quadratic equation: $l^2 - 17l + 72 = 0$ .
Factorize Quadratic Equation: Factorize the quadratic equation: l - 8)(l - 9) = 0\. So, \$l = 8 or $l = 9$ .
Find Valid Dimensions: If $l = 8$ , then $w = 17 - 8 = 9$ . If $l = 9$ , then $w = 17 - 9 = 8$ . Both pairs $(8, 9)$ and $(9, 8)$ are valid as dimensions.

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