Question

a straw is placed inside a rectangular box that is 7 inches by 10 inches by 10 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. 10 in. 10 in. 7 in.

Explanation:

Step1: Find the diagonal of the base

The base of the box is a rectangle with length \(l = 10\) inches and width \(w=7\) inches. Using the Pythagorean theorem \(d_{base}^2=l^{2}+w^{2}\), we have \(d_{base}^2 = 10^{2}+7^{2}=100 + 49=149\).

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

The height of the box \(h = 10\) inches. Using the three - dimensional Pythagorean theorem \(d_{space}^2=d_{base}^2+h^{2}\). Substitute \(d_{base}^2 = 149\) and \(h = 10\) (so \(h^{2}=100\)) into the formula: \(d_{space}^2=149 + 100=249\). Then \(d_{space}=\sqrt{249}=\sqrt{3\times83}\) (since \(249 = 3\times83\) and \(3\) and \(83\) are prime numbers).

Answer:

\(\sqrt{249}\) inches