Question

2. [-75 Points] DETAILS SCALCCC4 7.5.007.MI. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER The population of the world was about 5.3 billion in 1990. Birth rates in the 1990s ranged from 35 to 40 million per year and death rates ranged from 15 to 20 million per year. Let's assume that the carrying capacity for world population is 100 billion. (Assume that the difference in birth and death rates is 20 million/year 0.02 billion/year.) (a) Write the logistic differential equation for these data. (Because the initial population is small compared to the carrying capacity, you can take k to be an estimate of the initial relative growth rate.) dP dt , P in billions (b) Use the logistic model to estimate the world population in the year 2000. (Round the answer to two decimal places.) P= billion (c) Compare with the actual population of 6.1 billion. (Round the answer to one decimal place.) billion. (d) Use the logistic model to predict the world population in the years 2100 and 2500. (Round the answer to two decimal The difference is places.) P= billion in the year 2100 P= billion in the year 2500 Need Help? Read It Master It Talk to a Tutor

Answer 1

Sonn Population of the world was about 5.3 billion in 1990. Birth Million per year rates in the 1990s ranged from 35 to 40 Million Рел year and death rates ranged from is to 20 Let's assume that the carrying capacity for world population in loo billion (assume that the The difference in birth and death rates is 20 Million / year ) write the logistic differential equation These data ( because the is initial population is small compared the carrying capacity, you Can take be an estimate the initial relative growth rate , con 'K' to 2 Initial growth rate = Birth rate - Death rate ( 35 to 40 million year)-(15 to 20 Millionlse) 20 Million / year rate Initial relative growth initial Growth rate initial population 20x106 s. 38109 7 2x10 5.38109 26S we because Carrying capacity is is much larger than initial population will assume Ka initial Relative Growth rate the 265

Logistic differential equation + de - K(1-2 substitute ㅗ 265 [ Given that carrying capacity 100 billion) lis & de 245 (1-eve Р 265 dP dt P (9-) here 265 p is in billion dP dt 2 25 (P) (*) 265 (6) use to estimate the the logistic Model world population in solution of The yeare 2010 uns dP p² P 265 100P - p2 26 Soo at 265ou dP 26 Soo et loop-p2 [integrate both side.

I et te 26500 p2_1000 ttc -265oo (to Un l-p+100) - t In (P)) = - 265 [un (-P+10o) In (P)] = tte t ico Plt) 2 265 e t 265 a с + е Plo) 5.3 billions PLO) 5.3 billions Itcie ca 17.8679 100 then Plt) t 265 It 17.8 69 e world population in 20vo 5.49 billion 10 P(10) = (و۔ [lt 17.8 69 e Pz 5.49 billion

(C) compare with actual population The dioberence is 6.1 billion 0.61 billions Git 5.49 0.61 billions estimated value of population is less than but near The actual value 6.1 billion old) use logistic Model to predict the world population in The years 2100 and 25 pa 7.81 billion the year 2100 27.72 billion the year 2500 the population in year 21w will be P(110) 7.81 billions - ( 11 lis) x It 17.8679 e and year 2500 P(510) 100 570 27.72 billion + 17.8679 e