Find the dimensions of a rectangular lawn whose perimeter is $100$ feet and whose area is $600$ square feet

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Q. Find the dimensions of a rectangular lawn whose perimeter is

100

feet and whose area is

600

square feet

Define Perimeter Formula: Let $l$ be the length and $w$ be the width of the rectangular lawn. $\newline$ The perimeter of a rectangle is given by the formula $P = 2l + 2w$ .
Perimeter Equation: Given the perimeter of the rectangular lawn is $100$ feet, we can write the equation: $100 = 2l + 2w$
Simplify Perimeter: Simplify the perimeter equation by dividing all terms by $2$ : $50 = l + w$
Define Area Formula: The area of a rectangle is given by the formula $A = lw$ . $\newline$ Given the area of the rectangular lawn is $600$ square feet, we can write the equation: $\newline$ $600 = lw$
Area Equation: We now have a system of two equations with two variables: $\newline$ $50 = l + w$ (Equation $1$ ) $\newline$ $600 = lw$ (Equation $2$ ) $\newline$ We can solve this system by expressing one variable in terms of the other using Equation $1$ and then substituting into Equation $2$ .
Solve System of Equations: From Equation $1$ , express $w$ in terms of $l$ : $w = 50 - l$
Express $w$ in terms of $l$ : Substitute $w = 50 - l$ into Equation $2$ : $\newline$ $600 = l(50 - l)$
Substitute into Equation $2$ : Expand the equation and rearrange it into a quadratic equation: $\newline$ $600 = 50l - l^2$ $\newline$ $0 = l^2 - 50l + 600$
Expand and Rearrange: Factor the quadratic equation: $\newline$ $0 = (l - 20)(l - 30)$
Factor Quadratic Equation: Solve for $l$ by setting each factor equal to zero: $l - 20 = 0$ or $l - 30 = 0$ $l = 20$ or $l = 30$
Solve for $l$ : If $l = 20$ , then $w = 50 - l = 50 - 20 = 30$ . If $l = 30$ , then $w = 50 - l = 50 - 30 = 20$ . So, the length and width of the lawn can be either $(20, 30)$ or $(30, 20)$ feet.