To find the length of the diagonal piece of wood in a 12-inch square box, we can use the Pythagorean theorem. For a square, the diagonal \(d\) can be calculated using the formula:
\[ d = \sqrt{a^2 + a^2} \]
where \(a\) is the length of one side of the square. In this case, \(a = 12\) inches.
Substituting the value of \(a\):
\[ d = \sqrt{12^2 + 12^2} = \sqrt{144 + 144} = \sqrt{288} \]
Now, simplifying \(\sqrt{288}\):
\[ \sqrt{288} = \sqrt{144 \times 2} = \sqrt{144} \times \sqrt{2} = 12\sqrt{2} \]
Next, we approximate \(\sqrt{2}\) which is approximately \(1.414\):
\[ d \approx 12 \times 1.414 \approx 16.968 \]
Rounding to the nearest inch gives:
\[ d \approx 17 \text{ inches} \]
Therefore, the length of the diagonal piece of wood needed is approximately \(17\) inches.
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