vmt-award-2024_48-1-_red

→0 + . (4b)

The positive root of this equation is, after some rearrangement,

Therefore, k = C 0 / R = (4.8 × 10 8 ) / (2.5 × 10 11 ) ≈ 1.9 × 10 - 3 = 0.19% per year or 1.9% per decade.

. (6)

In the graphs below, we plot T ( a ) for several dif ferent values of k , and T ( k ) for different values of a , all using the above value for C 0 / R , which is, R / C 0 ≈ 520.8.

The linear - linear model

We can deviate from the exponential growth/decay model of course, and pick linear or other forms, and combine them as well. Before combining them, we will review this process using the linear linear model for pedagogical purposes. We could draw on data analysis and investigate the actual data for C ( t ) and attempt to fit a curve to the data, but that is outside the scope of this article. Let us assume that the consumption rate C ( t ) of a non renewable resource, quantity R , increases linearly with the slope k from a value C 0 at t = 0 in the in terval [0, aT ], where 0 < a <1, and from there de creases linearly to zero at t = T . For ease of read ing, we drop the subscript e . T is the depletion time when the resource R is completely exhausted. We can derive an explicit expression for T . The ad vantage of this is that since the area under the graph of C ( t ) from t = 0 to t = T must equal R , so we do not need to use calculus to integrate it to find the value, we use geometry to find the area under the curve to solve for T .

Figure 2 : Upper curve to lowest curve: k = 0.05; 0.1 and 0.2 respectively.

Figure 3 : Upper curve: a = 0.5; lower curve: a = 0.75.

These curves clearly illustrate the fact that increas ing the rate of consumption reduces the resource expiration time.

The exponential - linear model

Figure 1 : C(t) diagram for the linear - linear model.

Another modification to the basic model is to as sume exponential consumption of resources up to time aT , where 0 < a <1 as before, followed by a linear decrease until time T . The equation for linear decrease is established using the point - slope for mula, ie. . The area under the graph of C ( t ) can be divided into two portions: under the exponential growth until t = aT , and the triangular region from aT to T . This is simple to accomplish,

In figure 1 we sum the areas 1 + 2 + 3 = R . These areas are represented by two right triangles ( 1 and 3 ) and one rectangle ( 2 ) respectively.

Area 1 =

area 2 = C 0 aT and area

3 = Simplifying these expres sions produces the following quadratic equation in T: . (5)

Virginia Mathematics Teacher vol. 48, no. 1

40