Question

The larger square garden at Volterra Hall has sides twice as long as the smaller square garden. Together the gardens cover 18,000 square feet. Find the dimensions of each garden.

Answer 1

We will call:
$G_{1}$: The larger square garden.
$G_{2}$: The smaller square garden.

Given that both of then have square areas, then:
$A_{G1} = L_{G1}^{2}$
$A_{G2} = L_{G2}^{2}$

Being $L_{G1}$ the side of the larger garden and $L_{G2}$ 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:

$L_{G1} = 2L_{G2}$

Together the gardens cover 18,000 square feet, so:
$A_{G1} + A_{G2} = L_{G1}^{2} + L_{G2}^{2} = 18000$

Then:
$(2L_{G2})^{2} + L_{G2}^{2} = 18000$
$4L_{G2}^{2} + L_{G2}^{2} = 18000$
$5L_{G2}^{2} = 18000$
$L_{G2}^{2} = 3600$
$L_{G2} = \sqrt{3600}$
$L_{G2} = 60$
$L_{G1} = 2L_{G2} = 2(60) = 120$

Therefore:
Each side of the larger square garden is $120 ft$ and each side of the smaller one is $60 ft$ each. These are the dimensions.