Question

a straw is placed inside a rectangular box that is 3 inches by 3 inches by 8 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.

Explanation:

Step1: Find the diagonal of the base

The base is a square with side - length \(a = 3\) inches. Using the Pythagorean theorem \(d_{base}^2=a^{2}+a^{2}\) (where \(d_{base}\) is the diagonal of the base).

$$d_{base}^2=3^{2}+3^{2}=9 + 9=18$$

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

Let the height of the box \(h = 8\) inches. The length of the space - diagonal \(d\) of the rectangular box can be found using the formula \(d^{2}=d_{base}^2+h^{2}\).
We know \(d_{base}^2 = 18\) and \(h = 8\), so \(d^{2}=18+8^{2}\).

$$d^{2}=18 + 64=82$$

$$d=\sqrt{82}$$

Answer:

\(\sqrt{82}\)