Consider the following list of modeling tools and modeling situations. Match eac

Question

Consider the following list of modeling tools and modeling situations. Match each modeling situation with the appropriate modeling tool by writing the number of the appropriate tool next to each situation. Tools: (i) Markov chain (ii) linear interpolation (iii) second order differential equation (iv) first order separable differential equation (v) cubic spline interpolation (vi) line of regression (vii) R^2 (viii) first order recurrence relation. Situations: (a) You are given monthly data on the total population of sandhill cranes in a small portion of the Florida Everglades over a period of one year. You want a continuous model for the crane population which agrees with the observed monthly population and is also twice differentiable at any time strictly between zero and one year. (b) You are given monthly data on the total population of sandhill cranes in a small portion of the Florida Everglades over a period of one year. Based on the data, you want to estimate the crane population at the end of the following year. (c) You are given a function P = f(t) which is supposed to model the total crane population. You want to see how well the function fits the actual data. (d) You have a 30-year, $80,000 mortgage at 7.2% annual interest compounded monthly with a monthly payment of $543. You want to model the balance of your loan on a month by month basis. (e) A region of land has two types of coverage: grassland and forest. Over the period of any single year, there is an 80% chance that grassland will remain grassland, and there is a 60% chance that forest will remain forest. You want to study the long-term proportions of grassland and forest. (f) It has been observed that the temperature of an object changes at a rate proportional to the difference between the temperature of the object and the ambient temperature. You want to model this observation.

Explanation / Answer

Situation (a) => Second order differential equation ; as the model justifies inclusion of double differentiation with time.

Situation (b) => LIne of Regression : The month and sand hill population will give line of regression on that basis we will calculate population of sand hill next year.

Situation (c) = > R2 : Regression coefficient is always the best way to check how the given data is perfect fit on the given function or relationship.

Situation (d) => FIrst Order Recurrance Relation : on a monthly basis it can be calculated by First order recurrence relation.

Situation (e) => Markov Chain : Markov chain is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. So, to calculate long term proportions of grassland and forest Markov chain is needed.

Situation (f) => First Order Seperable diffwerential equation : As the relationship is the linear relationship.

i.e. dT/dt = k ( T - To) , so it is first order "separable" as temperature and time can be separated and integrated differently and differential equation.

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