The prominent attempt to forecast growth mathematically was made by the British economist and clergyman, Thomas Malthus, in the essay published in 1798. Malthus argued that growth of human population would overtake growth of food supplies, as population multiplied by the fixed amount each year but food supplies grew by adding the fixed amount each year. He accomplished that humans were doomed to breed to point of starvation unless population was reduced by other means, like war or disease.
The table is based on Malthus' computations with years from 1800. A food unit is defined to be enough food for one person for one year.
| Year |
0 |
5 |
10 |
15 |
20 |
25 |
30 |
35 |
| Population of England in millions |
7 |
8.04 |
9.23 |
10.60 |
12.16 |
13.96 |
16.03 |
18.4 |
| Food units in millions |
7 |
8.04 |
9.80 |
11.20 |
12.60 |
14.00 |
15.4 |
16.8 |
The equation p(t) = 7(1.148)^{t/5}models the population of England (in millions) as the function of time in years from 1800.
(a) Describe how numbers 7, 1.148 and 5 are associated to situation modeled by equation. Equation f (t) = 7+1.4(t/5) models number of food units produced each year (in millions) as the function of time in years sine 1800.
(b) Describe how numbers 7, 1.4 and 5 are associated to situation modeled by equation.
(c) Determine slope of equation and interpret slope in context of problem. Provide units for slope.
Model Malthus used predicts the point at which food production would just meet needs of population. That point may be modeled by setting Population = Food units produced
7(1.148)^t/5 = 7+1.4(t/5)
(d) Use table to find year when this happens
(e) Even with the knowledge of logarithms, system can't be solved using algebra. Illustrate why this is true by taking several algebraic steps to attempt to solve equation for t. What occurs when we try to use logarithms to get variable out of exponent? What goes wrong when trying to solve this way?