Given ratio of the width of Francois's wife's vegetable garden to its length is 5:8.
Let l1,w1 be the length and width of Francois's wife's vegetable garden.
Then [tex]\frac{w1}{l1} = \frac{5}{8}[/tex]
Given ratio of the width of the herb garden to its length is 3:5.
Let l2,w2 be the length and width of herb garden.
That is [tex]\frac{w2}{l2} = \frac{3}{5}[/tex]
Given that length of the herb garden is same as the width of the vegetable garden.
That is l2=w1 let this common value be x.
So, first ratio is [tex]\frac{x}{l1}=\frac{5}{8}[/tex]
l1 = [tex]\frac{8x}{5}[/tex]
Second ratio is [tex]\frac{w2}{x} = \frac{3}{5}[/tex]
w2 = [tex]\frac{3x}{5}[/tex]
perimeter of vegetable garden = 2(l1+w1) = [tex]2(\frac{8x}{5} +x) = 2*\frac{13x}{5} = \frac{26x}{5}[/tex]
Perimeter of herb garden = 2(l2+w2) = [tex]2(x+\frac{3x}{5} ) = 2*\frac{8x}{5} =\frac{16x}{5}[/tex]
Given that francois has 252ft of fencing material.
That is perimeter of vegetable garden + perimeter of herb garden = 252
[tex]\frac{26x}{5} +\frac{16x}{5} = 252[/tex]
[tex]\frac{42x}{5} =252[/tex]
[tex]x = \frac{252*5}{42} = 30 feet[/tex]
So, [tex]l1 = \frac{8x}{5} = \frac{8*30}{5} = 48 feet[/tex]
And [tex]w2 = \frac{3x}{5} = \frac{3*30}{5} = 18 feet[/tex]
So dimensions of vegetable garden are 48ft, 30ft.
And dimensions of herb garden are 30ft,18ft.
