Optimizing Pencils in a Tray
How many pieces of chalk can you fit along a chalkboard tray?Copyright 2016 Pixabay CC0 (public domain) https://pixabay.com/en/cartoon-chalkboard-education-female-1297710/
Student groups work with manipulatives—pencils and trays—to maximize various quantities of a system. They work through three linear optimization problems, each with different constraints. After arriving at a solution, they construct mathematical arguments for why their solutions are the best ones before attempting to maximize a different quantity. To conclude, students think of real-world and engineering space optimization examples—a frequently encountered situation in which the limitation is the amount of space available. It is suggested that students conduct this activity before the associated lesson, Linear Programming, although either order is acceptable.
Engineers sometimes need to optimize a system for a given quantity against one or more constraints. In this activity, the quantity to be maximized is the total length of all the pencils and the constraint is the tray length. Additionally, some engineering problems can be simplified if the spatial dimensions can be reduced from the standard three of our physical world to either two or one. While the pencils and the tray are themselves three-dimensional objects, the pencils can only fit stably inside by aligning their long axes with the tray’s long channel. Recognizing how a complicated problem can be made easier is an important skill for engineers!
After this activity, students should be able to:
- Pick the pencils from a pile that enable the greatest number of pencils to fit linearly (single file) in a tray.
- Pick the pencils from a pile that enable the greatest length to fit linearly (single file) in a tray.
- Pick the pencils from a pile that enable the greatest amount to fit linearly (single file) in a tray, given the additional requirement of fixed, nonzero gaps between pencils, including end pencils and walls.
- Explain how linear programming relates to space optimization in engineering design problems.