## Exponential Growth

Our energy dilemma can be analyzed in terms of fundamental principles. A corollary of the first law of thermodynamics is: it is a physical impossibility to have continued exponential growth of any product or exponential consumption of any resource in a finite system.

The present rate of consumption and the size of the system give a tendency for people to perceive the resource as either infinite or finite. The total energy output of the sun and the amount of mass in the solar system are infinite sources at our present rates of energy and material use, even though the solar system is finite. Even just the amount of solar energy received by the earth is a very large resource. The energy dilemma is defined within the context of the system, and our present energy dilemma is due to the finite amount of fossil fuels on the earth.

An easy way to understand exponential growth (Figure 2.1) is to use the example of money. Suppose Sheri receives a beginning salary of $1/year with the stipulation that the salary is doubled every year, a 100% growth rate. It is easy to calculate the salary by year (Table 2.1). After 30 years, her salary is $1,000 million per year. Notice that for any year, the amount needed for the next year is equal to the total sum for all the previous years plus 1.

Suppose a small growth rate is used, the doubling time (T2) can be estimated by

where R = % growth per unit time, generally years. Doubling times for some different yearly rates are given in Table 2.2.

FIGURE 2.1 Exponential growth with a growth rate of 100% per year.

TABLE 2.1

Salary at Growth Rate of 100% Per Year

TABLE 2.1

Salary at Growth Rate of 100% Per Year

Year |
Salary, $ |
Amount = 2' |
Cumulative, | ||||||||||||||||||||||||||

0 |
1 |
20 |
1 | ||||||||||||||||||||||||||

1 |
2 |
21 |
3 | ||||||||||||||||||||||||||

2 |
4 |
22 |
7 | ||||||||||||||||||||||||||

3 |
8 |
23 |
15 | ||||||||||||||||||||||||||

4 |
16 |
24 |
31 | ||||||||||||||||||||||||||

5 |
32 |
25 |
63 | ||||||||||||||||||||||||||

6 |
64 |
26 |
127 | ||||||||||||||||||||||||||

7 |
128 |
27 |
255 | ||||||||||||||||||||||||||

8 |
256 |
28 |
511 | ||||||||||||||||||||||||||

t |
2t |
2t+1 - 1 | |||||||||||||||||||||||||||

30 |
1 * 109 |
230 |
Doubling Times for Different Rates of Growth TABLE 2.2 Doubling Times for Different Rates of Growth
## Solar Panel BasicsGlobal warming is a huge problem which will significantly affect every country in the world. Many people all over the world are trying to do whatever they can to help combat the effects of global warming. One of the ways that people can fight global warming is to reduce their dependence on non-renewable energy sources like oil and petroleum based products. |

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