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Packaging
The solution from Bucharest
for the Dutch task
Exercise 1: The cuboid
S =2a(2h+a)
V=l*l*h=1000 cm3
Since the side and the height are integer numbers, we looked for the divisors of 1000 that are perfect squares. They are: 1, 4, 25, 100, and the correspondent height values are: 1000, 250, 40 and 10.
We calculated the surface area in each of these cases
In conclusion, the most cost-effective cuboid is the cube. The surface/volume ratio for the cube is 6/l, in our case S/V=0.6
Exercise 2: The cylinder
We aproximated the cylinder by a cuboid with the same height and the base the square circumscribed to the base of the cylinder. In order to be the most cost-effective, the cuboid must be a cube, as shown in the previous case, so h=a=2r
S/V=3/r=0.(5)<0.6
The cylinder is more cost-effective than the cube.
Exercise 3: The sphere
Therefore, the sphere is the best package, but it’s not practical for the store and the fridge. S/V=0.4838
Some conclusions:
Regarding the last question, if we consider the volume fixed, the answer is no. But, we found some interesting things. The S/V ratio depends on the dimension of the solid (side, radius), therefore :
1. this explains why powdered sugar dissolves faster than the regular sugar
2.the S/V ratio is very important in the living world, because it influences the impact of the surrounding enviroment on te living organism. The ratio is important for warm-blooded animals because the amount of heat lost by the body is proportional to its surface area, whereas the amount generated is proportional to its volume. Very small birds and mammals lose a lot of heat and need a high intake of food to maintain their body temperature. Elephants, on the other hand, are in danger of overheating, which is why they have no fur.
3. if we look into the following table http://en.wikipedia.org/wiki/Surface-area-to-volume_ratio, the expression for the dodecahedron seems smaller than the one for the sphere (2.694/a). However, a dodecahedron and a sphere of the same dimension will not have the same volume. I’m not so sure about this table anyway.