The key information here from which is good to start is that

whose perimeter is equal to that of a square whose area is 1

PR (perimeter rectangle) = PS (perimeter square)

The area of the square is 1. Now to have a square area = 1, one side of the square must be 1 as well. Area square is \(1^2=1\)

So its perimeter is \(1 \times 4 = 4\); \(PS = 4\)

The area of a rectangle is \(PR = \ell \times w\)

In our case, the length is twice its width; \(PR = \ell + 2w + 2we + \ell = 2\ell + 4w\)

\ell = w

we do have that PR = 6w

\(PR=PS >>>>>>> 4 = 6w >>> w=\frac{4}{6} = \frac{2}{3}\)

\(2w = \frac{4}{3}\)

Area rectangle \(\ell \times w = \frac{4}{3} \times \frac{2}{3} = \frac{8}{9}\)

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