Question

a rectangular box is 4 cm wide, 4 cm tall, and 6 cm long. what is the diameter of the smallest circular opening through which the box will fit? the diameter is \\(\square\\) cm. (round to the nearest tenth of a centimeter as needed.)

Explanation:

Step1: Identify the space diagonal

To find the smallest circular opening, we need the space diagonal of the rectangular box. The formula for the space diagonal \( d \) of a rectangular prism with length \( l \), width \( w \), and height \( h \) is \( d = \sqrt{l^2 + w^2 + h^2} \). Here, \( l = 6 \) cm, \( w = 4 \) cm, \( h = 4 \) cm.

Step2: Substitute values into the formula

Substitute \( l = 6 \), \( w = 4 \), \( h = 4 \) into the formula: \( d = \sqrt{6^2 + 4^2 + 4^2} = \sqrt{36 + 16 + 16} = \sqrt{68} \).

Step3: Calculate the value

\( \sqrt{68} \approx 8.246 \), rounded to the nearest tenth is \( 8.2 \).

Answer:

\( 8.2 \)