Mathematical Modeling Module
Alabama School of Mathematics and Science
1255 Dauphin Street, Mobile, AL 36604

Learning Modules/Mathematical Modeling/C.Akkoc/1996

MODULE No. 2

MAXIMIZING the VOLUME of a CARDBOARD BOX with a FIXED AMOUNT of MATERIAL

An open RECTANGULAR box is to have a SQUARE base and is to contain 88.844sq_cm of cardboard, bottom and four sides combined. What dimensions will yield the maximum volume for the box? The box will be made from a flat piece of cardboard.



  1. Write down the model for the volume V(H) as a function of H (see Figure). The model is not going to be a 2nd degree polynomial in H, therefore the graph is not going to be a parabola. During these calculations you will have derived an expression for X(H).

  2. Graph the mathematical model V(H), the volume function, together with the graph of X(H) on-the-same-screen on PLOT using an appropriate window. Move the cursor to the maximum point on the graph of V(H) (eyeballing) and record the coordinates of this maximum point. Then move the cursor straight-down to determine the coordinates of the corresponding point on the graph of X(H) directly below the maximum point for V(H). These two points on the two graphs completely determine the dimensions of that box with maximum volume as well as the value of Vmax itself. Record all this information by hand (pencil only) on the hard copy that you will make in the next step.

  3. Make a hard copy of the graph in step-2 on a printer. Label the two graphs by hand as V(H) and X(H). Mark the maximum point on V(H) and the corresponding point on X(H) as described above, together with the associated coordinates on the hard copy by hand.

  4. On a blank sheet of paper show key steps in your calculations in Part1 together with your results.

  5. Construct a cardboard model of the box with maximum volume using the optimal dimensions found in Parts 1-4 above.
 Mathematical Modeling Module
Alabama School of Mathematics and Science
1255 Dauphin Street, Mobile, AL 36604

Learning Modules/Mathematical Modeling/C.Akkoc/1996

MODULE No. 3

MAXIMIZING the LENGTH of a LADDER TURNING a 90 Degree CORNER

A hallway 5 distance units wide connects at 90 degrees to another hallway 2 distance units wide. Two men are carrying a ladder and the ladder has to turn the corner. The problem is to calculate the length of the longest ladder that will barely make the turn.



  1. Construct the mathematical model for the length of the ladder L(x) as a function of the distance "x" shown in the Figure. If L(x) happens to be a 2nd degree polynomial, then put it into the standard form by completing the square and determine Lmax algebraically. If the model turns out to be an expression other than a 2nd degree polynomial, we can not solve the problem algebraically, and the only recourse is to go to graphing software.

  2. In either case graph the mathematical model L(x), the length function, on PLOT using an appropriate window. Move the cursor to the minimum point on the graph of L(x) (eyeballing) and record the coordinates of this minimum point. What you have is the length of the LONGEST ladder that will make the turn.

  3. Make a hard copy of the graph in step-2 on a printer. Mark the maximum point on the graph of L(x) together with the associated coordinates on the hard copy by hand.

  4. On a blank sheet of paper show key steps in your calculations in Part1 together with your results.

  5. Construct a cardboard model of the hallway layout and a wooden model of the ladder to demonstrate that the maximum length found in Parts 1-4 above will barely make the turn.