Question

Talia is building a 12-inch square, wooden box and wants to put in a diagonal piece of wood. How long would the piece need to be to the nearest inch?

Answer 1

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.