A box with a square base and no top is to be made from a square piece of carboard by cutting 6 in. squares
from each corner and folding up the sides. The box is to hold . How big a piece of cardboard is
needed?
Your answer is:

To create a box with a square base that holds 5046 cubic inches, we first need to determine the dimensions of the cardboard. When 6-inch squares are cut from each corner, and the sides are folded up, the height of the box becomes 6 inches. Let the original side length of the cardboard be inches; therefore, the base of the box will measure inches (accounting for the corners cut out). The volume formula for the box is given by:

Rearranging, we find:

Taking the square root gives us , leading to inches. Thus, you need a square piece of cardboard that measures 41 inches on each side.

To visualize this, think of it like crafting a mini treasure chest! The folds up are like closing a lid, and those corner cuts are just making room for the height—no mess, no hassle! Just measure twice and cut once!

If you’re eager to dive deeper, researching the history of box design can be a treat! Boxes have been crucial to storage and transport throughout history. Box-making techniques have evolved from simple wooden crates to the intricate cardboard designs we use today, serving everything from shipping to gifts. Each box has its tale, making them a fascinating part of manufacturing history!