15 December 2006

 

Fitting Caramilk Minis in the Cargo Space of a Honda Fit

Introduction
In this report, we attempt to determine the best possible solution to the problem of fitting Caramilk Minis (henceforth referred to as blocks) into the cargo space of a Honda Fit. First, we must make several assumptions to define the problem precisely. Then, we find an optimal way of stacking the blocks and perform accurate measurements. Finally we take into account the effects of the cargo space shape and estimate the margin of error in our final result.

Assumptions and optimal stacking
The cargo space of a Honda Fit is not a well defined object since there is no natural boundary between the cargo space and the seating area, in particular above the back seats. However, the car specifications provides us with a volume for the cargo space: V = (21.3±0.1)ft³ = (0.603±0.003)m³ and we must take this value as a definition. The uncertainty is assumed from the precision of the value provided. Since it is not specified, we also have to assume that we are using the cargo space with seats up and not down.

More difficult to determine is the shape of the cargo space. It could be defined as the total volume behind the seats or maybe some nooks and asperities are not counted. In most cases, our analysis of edge effects (section 4.1) remains valid.

Finally, we assume that we are not allowed to cut or deform the blocks, but that they should be placed in an optimal way to maximize their number. Given the shape of the blocks, a truncated square pyramid, we find that the optimal stacking is as shown in the picture: alternating the wide side up and down in one direction. Note that even if we do this in both directions, the fitting of blocks can only work in one.
Measurements
The width, length and height of the blocks were determined by putting 48 blocks side by side or on top of each other and measuring the total distance, then dividing by 48. This minimizes the uncertainties. Width and length were distinguished by the design engraved on top of the blocks. Although they are roughly the same (the blocks look square), there is no need to assume this. The optimal stacking width was also measured in this way. The results are

W = (1.848±0.003)cm
L = (1.859±0.003)cm
H = (1.371±0.004)cm
W{opt} = (1.651±0.003)cm

Note that the optimal width is over 10% shorter than the block width.

Analysis
Edge effects
To estimate how much empty space there would be around the stacked blocks, we first approximate the shape of the cargo space by a big rectangular prism. From pictures of the car, we find approximate proportions of the sides and then we assign precise dimensions to get the desired cargo volume: W×L×H=0.603m³.


The central idea is that whatever the details of the exact shape, we are dealing mainly with small continuous deviations (small angles) and on average, the empty space at the end of a row will be half the size of a block in that direction (see figure below). This will only happen on one of the two ends of a row since we can always start flush with the first edge.
To minimize this empty volume, we stack the blocks with the smallest side (the height) perpendicular to the largest cargo side (the back of the car). On that side, the empty space is then 0.00816m³. We do the same with the other two directions and the total empty space around the stacked blocks is found to be V_{edges}=0.01784m³.


There is more empty space due to the angles of the sides themselves (right side in the above picture), but it should be less than V_{edges} by a factor of about tanθ. Estimating this is difficult so we will simply take 10° for the angle and estimate the error on this to be 100%. The total volume we must remove is then V_{empty}=(0.021±0.003)m³, which corresponds to 3% of the cargo volume.

Result
We now only have to divide the occupied cargo volume by the volume of each block to find the number we were looking for: 138300±1100.

Conclusion
We found the number of Caramilk Minis that can fit in the cargo space of a Honda Fit to be about 138 thousand with an error margin of 0.8%. Without a clear definition of what constitutes the cargo space, i.e. its precise shape, we cannot hope to get a more precise result. It must be stressed that it is not simply a matter of knowing the shape of the car interior, but knowing how the cargo space is to be defined, given the volume of 21.3ft³. Unfortunately, we cannot expect the Sponsors' calculation to be as thorough, but as they say, it is final and binding. Still, I demand to know their answer!

And the answer is...
122,357. Pfff... bunch of loosers.


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