Optimization Problem #18: Triangles, Trapezoid, Paper Crease
How to Use the GeoGebra Applet?
Please move slider a, which is a sub-segment for the bottom width of the paper.
Please observe that for values of a less than or more than 4.5,
we have lengths that are not of minimum value.
Please observe that when a = 4.5, we have the minimum length of the crease.
Please look at the graph of the function calculating the length of the crease (in pink).
At what point do we have a tangent slope of 0?
Hint: Locate slider a in the Algebra Pane and manually change a to 4.5.
Next, try to move angle slider v to rotate the trapezoid into a position where the bases become horizontal lines instead of vertical lines.
Hint: Locate slider v in the Algebra Pane and manually change the angle to 90 degrees.
Finally, try to move angle slider o to rotate the upper part of the trapezoid (after dividing the trapezoid's highest vertical height into two) in order to form a rectangle.
Hint: Locate slider o in the Algebra Pane and manually change the angle to 180 degrees.
Please answer the multiple-choice questions for a self-assessment.
Zero Tangent Slope
1. What does it mean when the tangent slope of a function at a point is zero?
Minimum length of a crease
2. When finding the minimum length of a crease in a geometric problem, at what point does the tangent slope of the length function equal zero?
Zero Tangent Slope for the Optimization Problems
3. In the context of optimization, why is finding where the tangent slope equals zero important?
Derivative of the crease length function
4. How can the derivative of the crease length function in the context of a folded rectangular piece of paper help determine the minimum length of the resulting crease?
Two Horizontal Parallel Lines
5. Why did we rotate the trapezoid before dividing it by half of its highest height?
Divide the trapezoid into two parts
6. Why did we divide the trapezoid into two parts?