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PARACHUTE APPLICATIONS

Applications of Exponential

Growth and Decay

### Real-World Applications: Exponential Growth of the WEB and Newton's Law of Cooling

The applications of the concepts of expoential growth and decay are quite numerous and can cover almost any real world situation. In the following we include two applications of real world implementations of these concepts.

# WEB Data

## sults

The web has grown very fast. In fact, the web has grown substantially faster than the Internet at large, as measured by number of hosts.

The rate of the web's growth has been and continues to be exponential, but is slowing in its rate of growth. For the second half of 1993, the Web had a doubling period of less than 3 months, and even today the doubling period is still under 6 months.

 Results Summary Month # of Web sites % .com sites Hosts* per Web server 6/93 130 1.5 13,000 (3,846) 12/93 623 4.6 3,475 (963) 6/94 2,738 13.5 1,095 (255) 12/94 10,022 18.3 451 (99) 6/95 23,500 31.3 270 (46) 1/96 100,000 50.0 94 (17) 6/96 230,000 (est) 68.0 41 1/97 650,000 (est) 62.6 NA

Host in the final column is defined as a listed hostname. The number in parentheses is using the number of hosts responding to ping. See the Internet Growth Summary for more details.

The charts below show the exponential shape of this data:

II.                 Exponential Decay: Newton's Law of Cooling

The following is quoted from a web site of Palmer: Preliminary remarks

An aluminum beam brought from the outside cold into a machine shop where the regular normal temperature is maintained warms up to the temperature of the surrounding air. A hot silver ingot immersed in water cools to the temperature of the surrounding water.

In situations like these, the rate at which an object's temperature is changing at any given time is approximately proportional to the difference between its temperature and the temperature of the surrounding medium. This observation is sometimes called the Newton's Law of Cooling, although, as in the case with the aluminum beam, it applies to warming as well.

An equation representing this law can be written as

where T is the temperature of the object at time t, Ts is the surrounding temperature; and T0 is the value of T at time zero.

 USNA Mathematics Department Comments to wdj@usna.edu