Which is better: short broad tank or tall thin one?
The Sunday Nation
Nairobi,
23 February 2014
Most people are able to conceptualise and manipulate linear relationships between quantities. For example, it is reasonable to expect that a 2,000-litre-tank will be twice as tall as a 1,000L-one.
But the manufacturer may find it easier to increase the diameter of the tank instead of the height. In that case, relationship between the volume capacity of the tank and its dimensions become a little more complicated.
The reason is that the volume is equal to the product (that is; multiplication) of the base area and the height. Thus to get twice the volume, one can either double the height or double the base area (NOT the diameter!)
It is important to note that even though a bigger the diameter yields a bigger area, these two quantities are NOT linearly related. That is, when one is doubled, the other does not become twice as big!
The connection between these two quantities is a square relationship. If the diameter is changed by a certain factor, the area changes by the square of that factor. For example, if the diameter doubles (factor of 2), the area quadruples (factor of 2-squared; that is, 4).
Conversely, to get double area, we need to increase the diameter by a factor of the square-root of 2; that is 1.414.
Now a cylindrical 1,000L-tank might be 1m in diameter and 1.27m tall. To make a 2,000L-one, the manufacturer can either double the height to 2.54m or double the base area. In the second option, the diameter needs to go up from 1m to 1.414m – NOT 2m!
The question is: which is better? The manufacturer might be interested in the design that requires less material to make. Assuming that the thickness of the walls of the two tanks is the same, then a calculation of the total surface area will help in comparing the quantities of material used.
A cylindrical tank has three distinct surfaces: two flat discs (the top and the base) and the curved side. The surface area of a 1m-diameter disc is about 7,850 square centimetres; thus the two discs come to 15,700sq-cm. The area of the curved part is simply the circumference multiplied by the height. For the 2.54m-tall tank comes to 79,756sq-cm.
Therefore, the tall 2,000L tank requires about 95,456sq-cm of material to construct.
For the short but broad design, the flat discs are 15,600sq-cm each making 31,200sq-cm altogether. The curved part comes to56,387sq-cm. Thus the total material required to make the whole tank is 87,587sq-cm.
Clearly, the short but broad design consumes less material than the tall and thin one. The difference is about 8%. Unfortunately, many customers will find it hard to believe that the shorter design is actually 2,000L.
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