To find the length of the straw that fits diagonally in the rectangular box, we can use the formula for the space diagonal \( d \) of a rectangular box. The formula is given by:
\[ d = \sqrt{l^2 + w^2 + h^2} \]
where:
- \( l \) is the length,
- \( w \) is the width,
- \( h \) is the height of the box.
In this problem, the dimensions of the box are:
- Length \( l = 9 \) inches,
- Width \( w = 5 \) inches,
- Height \( h = 10 \) inches.
Now substituting these values into the formula, we have:
\[ d = \sqrt{9^2 + 5^2 + 10^2} \]
Calculating each term:
\[ 9^2 = 81 \] \[ 5^2 = 25 \] \[ 10^2 = 100 \]
Adding these values together:
\[ 81 + 25 + 100 = 206 \]
Now, taking the square root gives:
\[ d = \sqrt{206} \]
Next, we check if \( \sqrt{206} \) can be simplified. We look for the prime factorization of 206:
\[ 206 = 2 \times 103 \]
Since both 2 and 103 are prime numbers, there are no perfect squares in the factorization. Thus, \( \sqrt{206} \) is already in its simplest form.
Finally, we conclude that the length of the straw is:
\[ \boxed{\sqrt{206}} \]