MATHS PORTFOLIO 2(MOHAMMED MAKDA)

MATHS PORTFOLIO 2(MOHAMMED MAKDA) - MATHS PORTFOLIO TYPE 2...

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MATHS PORTFOLIO TYPE – 2 IBDP YEAR – 2006-08 TOPIC:- CREATING A LOGISTIC MODEL DONE BY:- MOHAMMED WAHID MAKDA Candidate Code Number – 002767-039 Grade- 12 Chandra VGWS Mohammed Makda Page 1
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“ DESCRIPTION” “A geometric population growth model takes the form = r × where r is the growth factor and Is the population at year n . For example, if the population were to increase annually by 20%, the growth factor is r = 1.2, and this would lead to an exponential growth. If r = 1 the population of stable. A logistic model takes a similar form to the geometric, but the growth factor depends on the size of the population and is variable. The growth factor is often estimated as a linear function by taking estimates of the projected initial growth and the eventual limit.” 1) “A hydroelectric project is expected to create a large lake into which some fish are to be placed. A biologist estimates that if 10,000 fish were introduced into the lake, the population of fish would increase by 50% in the first year, but the long-term sustainable limit would be about 60,000. From the information above, write two ordered pairs in the form ( where = 60,000. Hence, determine the slope and equation of the linear growth factor in terms of .” Let us put the values for the above pairs = 10,000 ( as initial number of fishes ) = 1.5 ( as 50% stated above and hence from the description ) = 60,000 ( given ) = 1 ( as the population becomes stable ) Slope = = = = = Mohammed Makda Page 2
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Substituting the values, we get Slope( m ) = = (-1 × ) or - Equation of the linear growth ( y - ) = m × ( x - ) Let y = Let x = And let and be 10,000 and 1.5 respectively - 1.5 = - × ( - 10,000 ) = - + + 1.5 = - = ( ) 2) “Find the logistic function model for .” From the above information given in the description, we can substitute the values and hence give logistic function model. Also from the equation of the linear growth, the model can be created. = r × = ( ) × Mohammed Makda Page 3
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Verifying the above equation for different values for n . For n = 0, = ( ) × ( = 10000 ) = ( ) × 10000 = 15,000 For n = 1, = ( ) × ( = 15000 ) = ( ) × 15000 = 21,750 For n = 2, = ( ) × ( = 21750 ) = ( ) × 21750 = 30,069.37 3) “Using the model, determine the fish population over the next 20 years and show these values using a line graph.” Mohammed Makda Page 4
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Value of ‘ n’ Population Growth Factor 0 10,000 1.5 1 15,000 1.45 2 21,750 1.382 3 30,069.37 1.299 4 39,069.32 1.209 5 47,246.79 1.127 6 53,272.27 1.067 7 56,856.28 1.031 8 58,643.68 1.013 9 59,439.07 1.005 10 59,772.48 1.002 11 59,908.47 1.000 12 59,963.30 1.000 13 59,985.30 1.000 14 59,994.12 1.000 15 59,997.64 1.000 16 59,999.05 1.000 17 59,999.62 1.000 18 59,999.84 1.000 19 59,999.93 1.000 20 59,999.97 1.000 { Scale } { X – Axis = Number of Years = 1 unit = 5 years } { Y – Axis = Population = I unit = 10000 } Mohammed Makda Page 5
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From the above table we know that after 20 years the value does not exceed 60,000 but is in decimals and very close to 60,000. Also the graph shows that
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MATHS PORTFOLIO 2(MOHAMMED MAKDA) - MATHS PORTFOLIO TYPE 2...

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