## Designing Things

Problems and tasks from a variety of sources intended to illustrate the way mathematics arises in life and work.

Packaging. Design cardboard packages for copier toners, cameras, and perfume bottles that can be die-cut efficiently from a large stock, then folded to make the boxes.

Designing Envelopes. Study the relation between the size of a greeting card and the shape of the envelope when it is unfolded. What should be the dimensions of a card whose envelope will form a rectangle before folding--thus reducing waste if envelopes are cut from large sheets?

Making Boxes. A small business intends to market an up-scale coffee beverage that is to be sold in one pint glass bottles packaged in hexagonal boxes with interlocking bottoms and flap-down lids. Knowing the dimensions of the bottle, design the pattern of a flat cardboard cutout that, when folded and glued, would form a hexagonal box that would contain one bottle snugly. Then determine the optimal dimensions of a rectangular cardboard box that can hold the individual boxes for shipment.

Projectiles. Design a device that can accurately and reliably hurl a marble to a precise point on a target whose distance may vary from 5 to 15 ft away.