Question: Dimensional Analysis

Dimensions are the basic types of units. Dimensional analysis is the examinations of dimensions of variables. Relationship between variables can be an important indication of nature of mathematical formulas. Fundamental dimensions are time (T), mass (M), and length (L). Note that the word "dimension" has several other meaning such as in "three-dimensional space" or "the dimension of a baseball field.

Experiment

We will be examining perimeters and volumes of cans of different shapes.

The cylinder in the left is the shape of a typical soup can and the right one is in the shape of typical tuna can. The radius of the soup can is 3 cm and tuna can is 5 cm. The height of the soup can is 10 cm and the tuna can is 4 cm.

Notice that cylinder is consisted of two circles and a rectangle. The width of the rectangle is the circumference of the circle. Write an equation for the total surface area and volume for a cylindrical can of radius of r and height h.

Above table should give you an idea of how height and radius can affect the surfaces area and volume of a can. We can tabulate the table for infinitely many values of h and r to find the dimension of a can that would uses the minimum material to contain a given volume. This would be very time consuming. With an aid of Calculus, we can make a computation and find the optimal can dimension.

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