This project consists of a short description of Thomas Malthus, his Malthusian model based on his essay, “An Essay on the Principle of Population”, and behavior of exponential functions in general, with a specific example involving a German cockroach whose picture is shown above. Sumit Sarkar
This project consists of a short description of Thomas Malthus, his Malthusian model based on his essay, “An Essay on the Principle of Population”, and behavior of exponential functions in general, with a specific example involving a German cockroach whose picture is shown above. Cockroach Project
Cockroaches are a species that is said to have survived since the time of the dinosaurs. One explanation for their survival is not only their adaptability to various changing conditions, but their extremely rapid rate of reproduction. Like most patterns of reproduction, the reproductive rates of the cockroach can be mathematically modeled. In this case we will be using what is known as the Malthusian model, applied to a particular genus of cockroach for which the binomial nomenclature of this insect is Blatella germanica. Thomas Robert Malthus was born on February 14, 1766, and popularized the economic theory of rent. He was the sixth of seven children. He invented the Malthusian growth model, named after him after he wrote an essay called “An Essay on the Principle of Population”, which was one of the most influential books about population. He was most influential in political, social, economic and scientific thought. The Malthusian growth model is an exponential law and therefore predicts a geometric growth in populations. It was used to justify the social policies of classical liberalism, which came be to be known as Social Darwinism. We have one pair of German cockroaches, one male and the other female. The binomial nomenclature of this insect is Blatella germanica. In our model, every 50 days each female cockroach will give birth to 18 male and female cockroaches, totaling 36 cockroaches overall. We are also assuming that the original couple died after the female had given birth to its eggs. We will also assume that it take 4 cockroaches to cover one square inch of ground. The exponential equation that represents this is y=2*18x. In this equation there are 2 variables, x and y. x is the number of 50 day periods, because every 50 days the female cockroach gives birth. y is the population at the given time of x. The constant is 2 because the original population was two cockroaches. In 18x, the base is 18 and not 36 because only the female cockroaches give birth, and there are 18 males and 18 females every offspring. This is the graph of the function. Exponential functions and linear functions are very different. For starters, the general formula for and exponential function is y = a*bx, where a is the y-intercept, and b is the growth or decay rate. A linear function can be represented in many different formulas, and the one commonly used is y = mx+b, in which y represents it’s value corresponding to the y-axis with respective values of x and b, m is the slope, and b is the y-intercept. In linear functions, x increases at a constant rate. For example, each time you add 1 to y, x will increase by a set number. In exponential functions, instead of increasing by a set number, the x value is multiplied by a certain number. Her is an example comparing both: y = 2x, each time you add 1 to y, the x value increases by two. If you have y = 2x, each time you add 1 to y, the x value is multiplied by two. Logarithms are used to generate data based on the Malthusian equation. They are the standard technique for managing exponential equations, which cannot be solved using conventional algebra. Since the exponent itself is a variable in the equation, it is necessary to work backwards to the root of the exponent in order to solve the equation. This can be done by taking the log of both sides, then solving using...
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