Box optimization
Box optimization, in this context, is the process of determining the dimensions of a box that maximize its volume given a fixed surface area. It involves expressing the volume as a function of one variable using the surface constraint and applying derivatives to find the dimensions that produce the maximum volume.
Interactive chart
Write the size of the sides of the surface in boxes \(a\) and \(b\). Move the \(x\) slider to change the size of the cut squares, move the \(\alpha\) bar to assemble or disassemble the box.
Derivation of the maximizing function
The objective is to find the largest volume for a box without a lid from a sheet. The optimization calculation is as follows:
$$\mathrm{Volume} = \mathrm{side\ a}\ \times \mathrm{side\ b}\ \times \mathrm{height}$$
$$\begin{split}V\left( x \right) &= \left(a-2x\right)\left(b-2x\right)x \\ &= 4x^{3} - 2 \left( a+b \right)x^{2} + abx \end{split}$$
Derive the function
$$V^{\prime}\left( x \right) = 12x^{2} - 4\left( a + b \right)x + ab = 0$$
Use the general formula to solve quadratic equations
$$x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \\ \Rightarrow x = \frac{\left( 4\left( a + b \right) \right) \pm \sqrt{\left( 4\left( a + b \right) \right)^{2}-4 \left( 12 \right) \left( ab \right) }}{2\left(12\right)}$$
Solve the equation to find \( x \) of the box:
$$x = \frac{1}{6}\left( a + b - \sqrt{a^2 - ab + b^{2}} \right)$$