We are given a rectangular box with dimensions 10 inches by 4 inches by 4 inches. A straw fits exactly into the box diagonally from the bottom left corner to the top right back corner. We need to find the length of the straw and express the answer in simplest radical form.

1. Problem Description

We are given a rectangular box with dimensions 10 inches by 4 inches by 4 inches. A straw fits exactly into the box diagonally from the bottom left corner to the top right back corner. We need to find the length of the straw and express the answer in simplest radical form.

2. Solution Steps

To find the length of the straw, we can use the 3D Pythagorean theorem. Let the dimensions of the box be $l$, $w$, and $h$. The length of the straw, $d$, which is the diagonal of the box, is given by:

$d = \sqrt{l^2 + w^2 + h^2}$

In this problem, we have $l = 10$ inches, $w = 4$ inches, and $h = 4$ inches. Plugging these values into the formula, we get:

$d = \sqrt{10^2 + 4^2 + 4^2}$

$d = \sqrt{100 + 16 + 16}$

$d = \sqrt{132}$

Now, we simplify the radical:

$132 = 4 \times 33$

$d = \sqrt{4 \times 33}$

$d = \sqrt{4} \times \sqrt{33}$

$d = 2\sqrt{33}$

3. Final Answer

The length of the straw is $2\sqrt{33}$ inches.