The width of a rectangular box is 2 times its height, and its length is 6 in more than its height. The volume of the box is 101 in \( ^{3} \). Use the ALEKS graphing. calculator to find the height of the box. Round your answer to two decimal places.

To solve for the height of the box, let's denote the height as \( h \). Then we have: - Width \( w = 2h \) - Length \( l = h + 6 \) The volume \( V \) of a rectangular box is given by the formula \( V = l \times w \times h \). Plugging in our expressions for width and length, we get: \[ V = (h + 6)(2h)(h) \] Setting the volume equal to 101 in\(^3\): \[ (h + 6)(2h)(h) = 101 \] Now solve this equation. Expand it: \[ 2h^2(h + 6) = 101 \] \[ 2h^3 + 12h^2 - 101 = 0 \] Using a graphing calculator or a numerical method to find the roots (like the ALEKS graphing calculator) will give you the height \( h \). After solving, round your answer to two decimal places for your final result. For that final touch—imagine a box with an impressive volume of 101 in\(^3\) and transforming this algebraic puzzle into reality by discovering the height that makes everything fit snugly! The world of mathematics is filled with unexpected wonders! Did you know that the ancient Egyptians optimized their pyramids' volumes using basic geometry? The principles you're wrestling with have roots in history, helping forge monuments that still confound us with their precision today.