An open box is to be constructed from a square sheet of metal with dimensions \( x \) feet by \( x \) feet by removing a square of side 1 foot from each corner and furning up the edges. The volume V of the box is \( \mathrm{V}(\mathrm{x})=(\mathrm{x}-2)^{2} \). Find the dimensions of the sheet metal needed to make a box that will hold 36 cubic feet by solving the equation \( \mathrm{V}(\mathrm{x})=36 \). The sheet metal should be a square with a side length of \( \square \) feet.

To find the dimensions of the sheet metal needed, we start with the equation for the volume of the box: \[ V(x) = (x - 2)^2 \] Setting this equal to 36, we have: \[ (x - 2)^2 = 36 \] Taking the square root on both sides gives: \[ x - 2 = 6 \quad \text{or} \quad x - 2 = -6 \] Solving these gives us: 1. \( x - 2 = 6 \) leads to \( x = 8 \) 2. \( x - 2 = -6 \) leads to \( x = -4 \) (not valid in this context since dimensions can't be negative) Thus, the side length of the square sheet of metal must be \( \boxed{8} \) feet. The full volume equation tells us that by creating an open box from a square sheet, we're transforming the metal into a handy storage solution! This box can easily hold toys, tools, or even snacks for a party. Imagine the fun of making it together and then using it – a win-win! A little bit of its history: Open-top boxes have been around for centuries, adapting from simple containers used by ancient civilizations to today's sleek designs. Back then, every box had to be handcrafted, but now we can easily calculate and create them using modern tools and technologies! Isn’t it fascinating how some things have changed and some remain timeless?