vmt-award-2024_48-1-_red

because the first part is found in equation (3). All we need to do is replace T by aT . Then, including the right triangular area we have

= R . (7)

(

) +

Upon rearrangement, an implicit expression in T is suitable for graphical purposes.

=1+

(8)

Again, for R / C 0 ≈ 520.8, the illustrative choice of k = 0.1 when a = 0.5 and a = 0.75 in Figure 4, show the intersections of the exponential functions, the left - hand side of the equation (8) and the linear functions, the right - hand side of the equation (8). When a = 0.5 the resource expiration time T ≈ 61 years whereas when a = 0.75, it is, T ≈ 47 years. The reduction is a result of the longer period of exponential consumption before the linear decrease begins. If a is smaller, the expiration time will in crease. For example, if a = 0.2, T ≈ 113 years, which is not shown.

Figure 5 : Proportion of the original resource re maining (solid curve); Corresponding curve plus the additional source.

Decade

Proportion remaining

Decade

Proportion remaining after new discovery

1 2 3 4 5 6 7 8 9

0.99902 0.99707 0.99316 0.98534 0.96970 0.93842 0.87586 0.75073 0.50049

7

0.93382

8

0.86712

9

0.73372

10

0.46691

10.9

0.00477

10.9(0688)

0

10

0

cious thinking because as both the graph and the Table reveal, the resource quickly runs out as time continues. Using an example from Goren (2011), suppose that an additional resource equal to the original amount R is located after 70 years. In Fig ure 5, the dashed curve shows that complacency is clearly not justified as also shown by the numbers displayed in Table 2 because we see the resources run out 10 years later. In this article we conveyed the notion that not eve rything in life is linear. This means, effect changes are not always directly proportional to cause changes in a particular situation. Specifically, we chose to focus on the exponential functions as ex cellent examples of nonlinear behavior. The con Table 2 : Sustainable Growth Thinking, no new re source (right side). Sustainable Growth Thinking, new resource (left side). Conclusion

Figure 4 : Solutions of equation (8) when k = 0.1 and a = 0.5 (solid curves), and when a = 0.75 (dashed curves).

Sustainable growth for Fossil Fuels?

To illustrate the fallacy of “ sustainable growth ” thinking, we turn this problem on its head, by sup posing we know the value of T e ,100 years, for ex ample. In units of decades, T e = 10. At any time prior to this, the amount of resource remaining is the change in R or Δ R = R - C ( t ). Hereafter, the proportion of the original resource remaining is shown as follows:

(

) = 1 - (

). (9)

This function is shown in Figure 5. What is sur prising is that even after 50 years, approximately 97% of the original resource remains, perhaps leading to complacency (see Table 2). This is falla

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