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 \[ \boxed{\sqrt{109}} \text{ inches.} \]
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