e+Maths

• Topic: Analysis of a real problem that leads to an equation. • Materials and tools: whiteboard to diagram, word and spreadsheet, use the netbook for problems of everyday life. • Skills involved: know how to analyze the problem, be able to identify the unknown more than adequate; be able to translate the problem as an equation, be able to assess the acceptability of the solution by referring to the real context. instead of learning to determine the solution 2 ° translation • Topic: Analysis of a real problem that leads to an equation. • Materials and tools: whiteboard to diagram, word and spreadsheet, use the netbook for problems of everyday life. • Skills involved: know how to analyze the problem, be able to identify the unknown more than adequate; be able to translate the problem as a inequality equation, be able to assess the acceptability of the solution by referring to the real context. instead of learning to determine the solution

mathematical models A mathematical model is an object ,or concept, that is used to represent something else. in particular, in a mathematical model, the actual phenomenon that we want to investigate is represented by typical amount of mathematics: functions, equations, variables, etc. ... The mathematical model of a real-world phenomenon is a process of rationalization and abstraction that allows to analyse the problem, describe it objectively and formulate its simulations, using a universal symbolic language. The modelling process proceeds in stages, which create a dynamic interaction between real world and mathematical world. As an example of the mathematical model, starting from simple physical or biological phenomena with concrete evolution time, we consider the dynamics of populations. We want to determine how a population evolves in the future, with the current population known. The simplest is Malthusian population model, named in honor of Malthus, the founder of modern demography. It provides an exponential abundance of a population and is based on the fundamental hypothesis that the ability to survive and to reproduce of each individual are not influenced by the presence of other individuals of the same species. The simplest method to calculate the number of individuals in a population at time t is the balance equation: Nt +1 = Nt - Bt + Dt Nt = λ Nt-1 = λ t N0 where: Nt +1 is the number of individuals at time t +1 Nt is the number of individuals at time t Dt is the number of the dead between t and t +1 Bt is the number of individuals born at time t and survivors

MODEL Malthus It is described by the equation Nt +1 = Nt λ where λ is the rate of growth. If λ <1 the population is in decline; if λ> 1 the population is growing; λ = 1 if the state is stationary. Known the value of the parameter λ, in order to calculate the abundance of the population after t generations starting from its initial abundance, It is sufficient to iterate the equation for T times. N1 = λ N0 N2 = N1 λ = λ 2 N0 N3 = N2 = λ 3 λ N0 ......................... Nt = λ = λ t-1 Nt N0 An

Analisi di un problema reale che conduca ad una disequazione


 * Sapendo che i costi fissi mensili ammontano a € 2100 e che il costo del materiale per ogni pupazzo è di € 3,50, determina quanti pupazzi ||
 * devono essere prodotti perché il bilancio non vada in perdita. ||  ||   ||   ||   ||   ||   ||
 * Analisi del problema ||  ||   ||   ||   ||   ||   ||   ||   ||   ||
 * 7€= prezzo di vendita ||  ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
 * 2100€= spese fisse ||  ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
 * 3,5€= costo pupazzo ||  ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
 * Individuazione delle variabili ||  ||   ||   ||   ||   ||   ||   ||   ||
 * x= numero pupazzi ||  ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
 * y=7x ||  |||| y=7x > 3,5x + 2100 ||   ||   ||   ||   ||   ||   ||   ||   ||
 * Calcolo delle soluzioni ||  ||   ||   ||   ||   ||   ||   ||   ||   ||
 * 7x > 3,5x + 2100 ||  ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
 * 3,5x > 2100 ||  ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
 * x > 2100/3,5 ||  ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
 * x > 600 ||  ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
 * y=7x ||  ||   ||   |||| y=3,5x + 2100 ||   ||   ||   ||   ||   ||   ||
 * numero pupazzi ||  || ricavo ||   || spese ||   ||   ||   ||   ||   ||   ||   ||
 * 0 ||  || 0 ||   || 2100 ||   ||   ||   ||   ||   ||   ||   ||
 * 100 ||  || 700 ||   || 2450 ||   ||   ||   ||   ||   ||   ||   ||
 * 200 ||  || 1400 ||   || 2800 ||   ||   ||   ||   ||   ||   ||   ||
 * 300 ||  || 2100 ||   || 3150 ||   ||   ||   ||   ||   ||   ||   ||
 * 400 ||  || 2800 ||   || 3500 ||   ||   ||   ||   ||   ||   ||   ||
 * 500 ||  || 3500 ||   || 3850 ||   ||   ||   ||   ||   ||   ||   ||
 * 600 ||  || 4200 ||   || 4200 ||   ||   ||   ||   ||   ||   ||   ||
 * 700 ||  || 4900 ||   || 4550 ||   ||   ||   ||   ||   ||   ||   ||
 * 800 ||  || 5600 ||   || 4900 ||   ||   ||   ||   ||   ||   ||   ||
 * 900 ||  || 6300 ||   || 5250 ||   ||   ||   ||   ||   ||   ||   ||
 * 1000 ||  || 7000 ||   || 5600 ||   ||   ||   ||   ||   ||   ||   ||
 * y=7x ||  ||   ||   |||| y=3,5x + 2100 ||   ||   ||   ||   ||   ||   ||
 * numero pupazzi ||  || ricavo ||   || spese ||   ||   ||   ||   ||   ||   ||   ||
 * 0 ||  || 0 ||   || 2100 ||   ||   ||   ||   ||   ||   ||   ||
 * 100 ||  || 700 ||   || 2450 ||   ||   ||   ||   ||   ||   ||   ||
 * 200 ||  || 1400 ||   || 2800 ||   ||   ||   ||   ||   ||   ||   ||
 * 300 ||  || 2100 ||   || 3150 ||   ||   ||   ||   ||   ||   ||   ||
 * 400 ||  || 2800 ||   || 3500 ||   ||   ||   ||   ||   ||   ||   ||
 * 500 ||  || 3500 ||   || 3850 ||   ||   ||   ||   ||   ||   ||   ||
 * 600 ||  || 4200 ||   || 4200 ||   ||   ||   ||   ||   ||   ||   ||
 * 700 ||  || 4900 ||   || 4550 ||   ||   ||   ||   ||   ||   ||   ||
 * 800 ||  || 5600 ||   || 4900 ||   ||   ||   ||   ||   ||   ||   ||
 * 900 ||  || 6300 ||   || 5250 ||   ||   ||   ||   ||   ||   ||   ||
 * 1000 ||  || 7000 ||   || 5600 ||   ||   ||   ||   ||   ||   ||   ||
 * 900 ||  || 6300 ||   || 5250 ||   ||   ||   ||   ||   ||   ||   ||
 * 1000 ||  || 7000 ||   || 5600 ||   ||   ||   ||   ||   ||   ||   ||