Maximum Volume of a Box 

Copyright Maplesoft, a division of Waterloo Maple Inc., 2007 

Introduction 

This application is one of a collection of examples teaching Calculus with Maple. These applications use Clickable Calculus? methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips.  Click on the buttons to watch the videos. 

 

The steps in the document can be repeated to solve similar problems. 

Problem Statement 

Determine the dimensions of a lidless box of maximal volume that can be formed from a sheet of 20 cm by 30 cm cardboard by cutting equal squares from the corners and folding up the sides.  See Figure 1.  

 

Solution 

 

 

Figure 1: Schematic for fashioning box

 

Figure 1 provides the basis for the solution of this problem.  If and are the length, width, and height respectively of a box, its volume is given by the formula
 

 

(3.1)

 

 

 

From Figure 1, the dimensions of the box are constrained by the equations 

 

 

(3.2)

 

 

 

(3.3)

 

 

 

 

Using the constraint equations, determine expressions for the variables and .

Insert the limiting equation for the length by pressing [Ctrl][L] and typing in the label reference for the equation.  Right-click, choose Solve > Obtain Solution for > x. Repeat this process for the width, this time solving for y.

 

 

(3.4)

 

 

(3.5)

 

 

 

(3.6)

 

 

(3.7)

 

 

 

Substitute the expressions for and into the expression V thereby obtaining .

Construct an expression for the volume of the resulting box by inserting the label references for x and y.

 

 

(3.8)

 

 

Plot the expression for to estimate its maximum value

Right-click on the equation defining V=V(h) and select Right hand side.  Then right-click on the result and choose Plots>Plot Builder.  Within the plot builder change the range to go from 0 to 10, keep the other default settings and click Plot.

 

 

 

(3.9)

 

 

 

 

 

From the plot, it looks like is approximately when is approximately .  

 

Determine analytically by the following steps.

On a new line, insert the label reference corresponding to the right hand side of the equation V=V(h) and press [Enter].

 

 

(3.10)

 

 

Differentiate and simplify the expression for

Right-click on the expression and and choose Differentiate > h.  Then right-click on the derivative  and select Simplify

 

 

 

(3.11)

 

 

(3.12)

 

 

Now solve for h

Right-click on the simplified expression and select Solve > Numerically Solve

 

(3.13)

 

 

The larger value of h violates the constraint of equation 3.3    Substitute the smaller value for h into equation 3.2 and 3.3 to determine the value of and

Insert the label reference for the length constraint equation. Press [Enter].  Copy the smaller value for h, right-click on the constraint equation and select Evaluate at a Point, paste the point in the 'h=' field.  Solve for x by Right-clicking on the constraint equation evaluated at a point and choose Solve > Obtain Solution for > x.  Repeat for the width constraint equation.

 

(3.14)

 

 

(3.15)

 

 

(3.16)

 

 

 

(3.17)

 

 

(3.18)

 

 

(3.19)

 

 

Substitute the smaller value for into the equation to determine the  maximal volume of the lidless box.

Copy the smaller value for h.  Insert the equation label for the equation V=V(h).  Press [Enter].  Right click on the equation and select Evaluate at a Point, paste the point in the 'h=' field.

 

(3.20)

 

 

(3.21)

 

The dimensions that produce a maximum volume of from a sheet of by cardboard are , , for the length, width and height respectively. 

Legal Notice: The copyright for this application is owned by Maplesoft. The application is intended to demonstrate the use of Maple to solve a particular problem. It has been made available for product evaluation purposes only and may not be used in any other context without the express permission of Maplesoft.