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The area of a rectangular bathroom mirror is $20$ square feet. The perimeter is $24$ feet. What are the dimensions of the mirror? $\newline$ $\_\_\_$ feet by $\_\_\_$ feet

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Area and perimeter: word problems

Full solution

Q. The area of a rectangular bathroom mirror is

20

square feet. The perimeter is

24

feet. What are the dimensions of the mirror?

\newline

\_\_\_

feet by

\_\_\_

feet

Area Equation: Let's denote the length of the mirror as $L$ feet and the width as $W$ feet. We know that the area ( $A$ ) of a rectangle is given by the formula $A = L \times W$ . We are given that the area is $20$ square feet. $\newline$ So, we have the equation: $\newline$ $L \times W = 20$
Perimeter Equation: We also know that the perimeter $P$ of a rectangle is given by the formula $P = 2 \times (L + W)$ . We are given that the perimeter is $24$ feet. $\newline$ So, we have the equation: $\newline$ $2 \times (L + W) = 24$
Simplify Perimeter: Let's simplify the perimeter equation to express one variable in terms of the other. We can divide both sides of the perimeter equation by $2$ to get: $\newline$ $L + W = 12$
Express Width in Terms of Length: Now, we can express $W$ in terms of $L$ using the perimeter equation: $\newline$ $W = 12 - L$
Substitute Width into Area Equation: We can substitute $W$ from the perimeter equation into the area equation to find $L$ :
$L \times (12 - L) = 20$
Expanding this, we get:
$12L - L^2 = 20$
Rearrange Quadratic Equation: To solve for $L$ , we need to rearrange the equation into a standard quadratic form: $\newline$ $L^2 - 12L + 20 = 0$
Factor Quadratic Equation: Now, we can factor the quadratic equation: L - \(10)(L - $2$ ) = $0$ \
Solve for Length: Setting each factor equal to zero gives us the possible values for $L$ : $L - 10 = 0$ or $L - 2 = 0$ So, $L = 10$ or $L = 2$
Calculate Width: If $L = 10$ , then $W = 12 - L = 12 - 10 = 2$ . If $L = 2$ , then $W = 12 - L = 12 - 2 = 10$ . Since a rectangle's dimensions are interchangeable, we have two possible sets of dimensions for the mirror: $10$ feet by $2$ feet or $2$ feet by $10$ feet.

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