Student handout: The Park

Surfaces/Bridge Workshop 2023

This is the first activity relating the surfaces to the corresponding contour diagrams, thus emphasizing the use of multiple representations.

Students work in small groups to interpret level curves representing different concentrations of lead.

What students learn
  • Level curves can be used to represent functions of two variables;
  • How to translate between surfaces and level curves;
  • The use of multiple representations can enhance students' ability to interpret their results.

On your Mark: You work for Granite Falls, a town which needs to move a playground due to harmful levels of lead (Pb) in the ground. The surface's height represents the concentration of lead (in \(\frac{\mathrm{g}}{\mathrm{m}^2}\)) in the topsoil at every location in Granite Falls. Lead levels range from \(0.5 \frac{\mathrm{g}}{\mathrm{m}^2}\) to \(6\frac{g}{m^2}\).

  1. The park is currently located at the red star. Mark all points on the surface with the same concentration of lead as the park.
  2. Mark all points with lead concentrations \(1 \frac{\mathrm{g}}{\mathrm{m}^2}\) higher and \(1 \frac{\mathrm{g}}{\mathrm{m}^2}\) lower than at the park.
  3. Could these curves intersect? Why or why not?
  4. Sketch the curves by projecting them down onto a sheet of paper. Give appropriate labels.

Get Set: Granite Falls is a \(10~\mathrm{km} \times 10~\mathrm{km}\) town in the \(15.5~\mathrm{km}\times 21.5~\mathrm{km}\) Rock County. Find where the town is located in the county and place it there. Explain how you know you found the right location.

Go: The park measures \(20~\mathrm{m} \times 20~\mathrm{m}\) at the red star. Determine the total mass of lead at the current location. Give units.

Challenge: The town wants to guarantee the total amount of lead in the park is less than \(600~\mathrm{g}\). Find a location near the red star that achieves this level. Explain how you found this new location.

Copyright 2014 by The Raising Calculus Group

Aaron Wangberg and the Raising Calculus team; used by permission.
Learning Outcomes