Question

a straw is placed inside a rectangular box that is 10 inches by 4 inches by 4 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 attempt 1 out of 2
in. submit answer

Explanation:

Step1: Find the diagonal of the base

The base of the box is a rectangle with length \(l = 10\) inches and width \(w=4\) inches. Using the Pythagorean theorem \(d_{base}^2=l^{2}+w^{2}\), we have \(d_{base}^2 = 10^{2}+4^{2}=100 + 16=116\).

Step2: Find the space - diagonal (length of the straw)

The height of the box \(h = 4\) inches. The space - diagonal \(d\) of the rectangular box is found using the formula \(d^{2}=d_{base}^{2}+h^{2}\). Substitute \(d_{base}^2 = 116\) and \(h = 4\) (so \(h^{2}=16\)) into the formula: \(d^{2}=116+16\). Then \(d^{2}=132\). Factor \(132\) as \(132=4\times33\). So \(d=\sqrt{4\times33}\). Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (\(a = 4\), \(b = 33\)), we get \(d = 2\sqrt{33}\) inches.

Answer:

\(2\sqrt{33}\)