Question

an architect designs a rectangular flower garden such that the width is exactly two - thirds of the length. if 420 feet of antique picket fencing are to be used to enclose the garden, find the dimensions of the garden. what is the length of the garden? the length of the garden is

Explanation:

Step1: Let the length of the garden be $l$ feet and the width be $w$ feet.

Given $w=\frac{2}{3}l$.

Step2: Recall the perimeter formula for a rectangle.

The perimeter $P = 2(l + w)$. Since $P=420$ feet and $w=\frac{2}{3}l$, we substitute $w$ into the perimeter formula: $420 = 2(l+\frac{2}{3}l)$.

Step3: Simplify the right - hand side of the equation.

First, combine like terms inside the parentheses: $l+\frac{2}{3}l=\frac{3l + 2l}{3}=\frac{5l}{3}$. Then $2(l+\frac{2}{3}l)=2\times\frac{5l}{3}=\frac{10l}{3}$. So the equation becomes $420=\frac{10l}{3}$.

Step4: Solve for $l$.

Multiply both sides of the equation by $\frac{3}{10}$: $l = 420\times\frac{3}{10}=126$ feet.

Step5: Solve for $w$.

Since $w=\frac{2}{3}l$, substitute $l = 126$ into the equation: $w=\frac{2}{3}\times126 = 84$ feet.

Answer:

The length of the garden is 126 feet and the width is 84 feet.