In many phenomena, quantities grow or decay at a rate proportional to their size.If the number of individuals in a population of animals or bacteria at time tisy=y(t), then the rate of growth dydtis proportional to the population y(t) for someconstant k.dydt=kyThis equation is sometimes called the law of natural growth (k>0)k<0y(t) given that y(0)=y0.(b) A certain culture of bacteria grows at a rate proportional to its size. If thesize doubles in4 days, find the time required for the culture to increase to10 times to its original size.ans: y(t)=y0ekt,13.3 daysUse the fact that the world population was 2560 million in1950 and 3040 millionin1960to model the population of the world in the second half of the 20th century.Assume that the growth rate is proportional to the population size.(a) What is the value ofk?(b) Use the model to estimate the world population in1993 and to predict thepopulation in the year 2020.ans: ,0.01718,5360 million, 8525 million