Box - Folding for Maximum Volume
Author: Linda Fahlberg-Stojanovska

How can we make a box with maximum volume from a given size "blank" of material?

You are given a "blank" of cardboard and asked to make a box out of it, that has the most possible volume.  You make the box by cutting squares of side [math]x[/math] from the corners, then folding the "flaps" up to form sides.  (The top stays open).

Leave the two check boxes unchecked to begin.

Set the desired size of your blank using the "Blank Length" and "Blank Width" sliders.  Then try different sizes of square cutouts by changing the "[math]x[/math]" slider.  Fold and unfold the side flaps using the "Box Open and Close" slider.  If necessary, adjust "Angle" and "Zoom" to get the best view of the box in the right-hand viewing window.

Remember that the volume of a rectangular prism is calculated as V = L W H.  Notice that each of the three dimensions of the box will change with [math]x[/math], so we might expect the volume to depend on [math]x[/math] as well.  L is NOT the blank's length; it is the length of the box after the corners have been removed.  Same with H.  Can you write a formula for each of L, H, and W in terms of [math]x[/math]?  If so, simply multiply them together to get [math]V(x)[/math], the Volume function.  Check the "Show Vol Calculation" box to check your answer.

FInding the exact value of [math]x[/math] that maximizes the volume requires calculus.  However, you can make a good estimate by finding the maximum value on the graph of [math]V(x)[/math].  You can display the graph by checking the "Show Vol Graph" box.  The actual value of [math]V[/math] for the current value of [math]x[/math] is plotted as a point, and the "optimum" value of [math]x[/math] (the value that gives the maximum volume) is plotted on the [math]x[/math]-axis.

Think about this question:  Part of [math]V(x)[/math] is highlighted in red; the rest is displayed as a gray dashed curve.  What is the significance of the highlighting?  Think about what the possible values of [math]x[/math] could be in the "real world".  This is called "domain restriction".  Often times, our mathematical models realistically apply only for a certain sub-domain of the function we are using.

x = %v

Blank Width = %v

Blank Length = %v

Zoom

Box Open/Close

Angle

x

Show Box

Show Vol Calculation

Show Vol Graph

x \space è  \space il \space lato \space del \space quadratino \space ritagliato

V \space è \space il \space volume \space della \space scatola

x

IL PROBLEMA DELLA SCATOLA

h = 0 (slider from 0 to 10, step 0.1)
width = 20 (slider from 1 to 20, step 0.1)
length = 19.9 (slider from 1 to 20, step 0.1)
Vright: Min[length, width] / 2
widBox: width - (2 * h)
lenBox: length - (2 * h)
V(x) = If[0 ≤ x ≤ h, ((x * (length - (2 * x))) * (width - (2 * x)))]
vol: V(h)
scale = 1.2 (slider from 1 to 3, step 0.1)
A = (4, 2)
γ = 1.5707963267948966 rad (slider from 0 to 1.5707963267948966, step 0.017453292519943295)
α = 0.4363323129985824 rad (slider from 0 to 1.5707963267948966, step 0.017453292519943295)
δp: γ + (α * (1 - (2 * γ / 3.14159)))
x1: ((scale * h) * cos(α))
y1: ((scale * h) * sin(α))
M: (x1, y1)
yC: (width * sin(α))
xC: (width * cos(α))
B: A + ((scale * length), 0)
C: B + (scale * (xC, yC))
b = Segment(B, C)
B' = Rotate(B, α, A)
s = Line(B', A)
Ax: A + (scale * (h, 0))
s1 = Line(Ax, s)
a = Segment(A, B)
r = Line(C, a)
D = Intersect(s, r)
Dy: D - M
r1 = Line(Dy, r)
Cy = Intersect(r1, b)
c1 = Segment(Dy, Cy)
c = Segment(C, D)
Dx = Intersect(s1, c)
d1 = Segment(Ax, Dx)
D1 = Intersect(d1, c1)
Bx: B - (scale * (h, 0))
q = Line(B, s)
q1 = Line(Bx, q)
Cx = Intersect(q1, c)
b1 = Segment(Bx, Cx)
C1 = Intersect(b1, c1)
δm: γ - (α * (1 - (2 * γ / 3.141592653589793)))
C2c: C1 + (scale * ((h * cos(δp)), (h * sin(δp))))
Ay: A + M
p = Line(A, B)
p1 = Line(Ay, p)
By = Intersect(p1, b)
a1 = Segment(Ay, By)
A1 = Intersect(d1, a1)
A2d: A1 + (scale * (((-h) * cos(γ)), (h * sin(γ))))
D2d: D1 + (scale * (((-h) * cos(γ)), (h * sin(γ))))
C2b: C1 + (scale * ((h * cos(γ)), (h * sin(γ))))
B1 = Intersect(b1, a1)
B2b: B1 + (scale * ((h * cos(γ)), (h * sin(γ))))
B2a: B1 + (scale * (((-h) * cos(δm)), (h * sin(δm))))
β = Angle(B, A, B')
d = Segment(D, A)
A2a: A1 + (scale * (((-h) * cos(δm)), (h * sin(δm))))
poly1, e, f, g, h_1 = Polygon(A1, A2a, B2a, B1)
poly2, i, j, k, l = Polygon(B1, B2b, C2b, C1)
D2c: D1 + (scale * ((h * cos(δp)), (h * sin(δp))))
poly3, c_1, d_1, e_1, f_1 = Polygon(C1, C2c, D2c, D1)
poly4, m, n, a_1, b_1 = Polygon(D1, D2d, A2d, A1)
Vder = Derivative(V)
chkShowBox = true
P: (h, V(h))
chkShowVCalc = false
chkShowVGraph = true
txtV: "Volume \,=\, \fgcolor{0000FF}{L}\,\cdot \, \fgcolor{FF0000}{W} \cdot \, \fgcolor{990099}{H}  \\
=(\fgcolor{0000FF}{" + (LaTeX[length]) + "}-2\fgcolor{990099}{x})(\fgcolor{FF0000}{" + (LaTeX[width]) + "}-2\fgcolor{990099}{x})(\fgcolor{990099}{x})\\
=\fgcolor{0000FF}{" + (LaTeX[lenBox]) + "}\, \cdot \, \fgcolor{FF0000}{" + (LaTeX[widBox]) + "}\, \cdot \, \fgcolor{990099}{" + (LaTeX[h]) + "} \,\\
=\, " + (LaTeX[vol]) + ""
E = Intersect(b1, a1)
F = Intersect(b1, a)
t = Segment(E, F)