- Visualisation of the Problem
The applet shows the cardboard and the slider at initial position. By moving the slider the box appears changing its shape. We try to guess the value of x where the volume is maximal and maximal volume of the box. - Table
The slider is at initial position, the point Vol (x, y) at origin. By moving the slider the point Vol(x,y) moves in the coordinate system, its coordinates filling the table . From the table we can read the value of x and maximal V(x) (the values are approximate). It will be interesting to compare the result to our guess in section 1. (Many students are really week at estimating results.) - Graph
The graph can be obtained in two ways: either directly by moving the slider (in this case the table should be disabled) or by plotting the points that correspond to the table (as we normally do in the classroom). Reading the coordinates of the highest point we get x and maximal V(x) - (Partly) Analytical Solution
V(x) can be expressed as a polynomial of 3. degree. Writing its equation into GeoGebra`s Input Bar its graph will appear. If our equation is correct, the points from section 3 will lie on it. By using the command Extreme the coordinates of both extreme will appear in the algebra window
Discussion:
What about minimum? Try to find some other, similar problem.
While the students look at the similar problems, they will generalize the problem and the solution procedure and look for different areas of application of the gained knowledge.