Question

A straw is placed inside a rectangular box that is 6 inches by 3 inches by 8 inches, as shown. If the straw fits exactly into the box diagonally from the bottom left corner to the top right back corner, how long is the straw? Leave your answer in simplest radical form.

Answer 1

To find the length of the straw that fits diagonally from the bottom left corner to the top right back corner of the box, we can use the three-dimensional distance formula.

First, we identify the dimensions of the rectangular box:
- Length (\(l\)) = 6 inches
- Width (\(w\)) = 3 inches
- Height (\(h\)) = 8 inches

The formula for the diagonal \(d\) of a rectangular box is given by:
\[
d = \sqrt{l^2 + w^2 + h^2}
\]

Now, we need to calculate each term:
- \(l^2 = 6^2 = 36\)
- \(w^2 = 3^2 = 9\)
- \(h^2 = 8^2 = 64\)

Next, we sum these squares:
\[
l^2 + w^2 + h^2 = 36 + 9 + 64 = 109
\]

Now we can find the length of the diagonal (and thus the length of the straw) by taking the square root:
\[
d = \sqrt{109}
\]

Therefore, the length of the straw is
\[
\boxed{\sqrt{109}} \text{ inches.}
\]