An example of a loaf of bread is as the figure below:
Figure 1: A Loaf of Bread (source: http://www.wikihow.com/Slice-Bread)
I am going to calculate the volume of a loaf of bread in the figure above. First of all, the cross-section of the loaf is as shown in the Figure 2 below:
Figure 2: Cross-section of the loaf of bread.
The cross-section of the loaf can be viewed as a combination of half of an ellipse and a rectangle, as in Figure 3 below:
Figure 3: Cross-section of the loaf as a combination of half of an ellipse and a rectangle.
A=area of12an ellipse+area of rectangleA=πS2r+h2r -----(1)
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where S is the the length of the semi-major axis of the ellipse, 2r is the length of the minor axis of the ellipse, and the area of the half of the ellipse is πS2r (source: http://www.mathsisfun.com/geometry/ellipse.html).
Figure 4: Cut the loaf into small slabs with length dx.
So,
The volume of one slab from the loaf=A dxA dx=2r πS+h dx
The length S is not a constant, and actually seems to follow an arc along the whole length of the loaf. Suppose that arc is from a parabola with the quadratic equation:
y=x-0-x+L= -x2+Lx,
where L is the total length of the loaf. Note that to come up with the quadratic equation, I have put one corner of the loaf on x = 0, and the other end of the loaf to be on x = L, so, we have the roots of the equation to be x = 0 and L. So, I can either get
y = (x-0)(x-L) or (x-0)(-x+L) and both still give the roots to be x = 0 and L. But when both are plotted, the second quadratic formula gives me a peak, and the first formula gives me an indentation. The peak is what I want because the loaf has a peak height, so I choose the second formula: y = (x – 0)( -x + L ).
Thus, the volume of one slab is:
A dx=2r πy-h+h dx=2r πy+h1-π dx
Integrating to find the volume of the loaf:
V= 0L2r πy+h1-π dx= 0L2r -πx2+πLx+h1-πdxV=2rπ0L-x2+Lx dx +2rh1-π0L1dxV=2rπ-13x3+L2x20L+2rh1-πL-0V=2rπ-13L3+L32+2rh1-πL Volume of a loaf of bread, V= 2rπ6L3+h1-πL
So, if we calculate for a typical loaf of bread ( like those in this source: http://www.kingarthurflour.com/shop/items/bread-loaf-pan-9-x-4-x-4 )
with a length of 9 inches, a height of 4 inches and a width of 4 inches, we get this information:
L=9, h=4, 2r=4,=>Volume, V=4π693+41-π*9≈1.22 X 103 inches3
