We will call:
[tex] G_{1}[/tex]: The larger square garden.
[tex] G_{2}[/tex]: The smaller square garden.
Given that both of then have square areas, then:
[tex] A_{G1} = L_{G1}^{2} [/tex]
[tex] A_{G2} = L_{G2}^{2} [/tex]
Being [tex] L_{G1} [/tex] the side of the larger garden and [tex] L_{G2} [/tex] the side of the smaller garden as shown in the figure.
The larger square garden at Volterra Hall has sides twice as long as the smaller square garden, thus:
[tex] L_{G1} = 2L_{G2}[/tex]
Together the gardens cover 18,000 square feet, so:
[tex]A_{G1} + A_{G2} = L_{G1}^{2} + L_{G2}^{2} = 18000[/tex]
Then:
[tex](2L_{G2})^{2} + L_{G2}^{2} = 18000[/tex]
[tex]4L_{G2}^{2} + L_{G2}^{2} = 18000[/tex]
[tex]5L_{G2}^{2} = 18000[/tex]
[tex]L_{G2}^{2} = 3600[/tex]
[tex]L_{G2} = \sqrt{3600}[/tex]
[tex]L_{G2} = 60[/tex]
[tex]L_{G1} = 2L_{G2} = 2(60) = 120[/tex]
Therefore:
Each side of the larger square garden is [tex] 120 ft [/tex] and each side of the smaller one is [tex] 60 ft[/tex] each. These are the dimensions.
