In addition to analyzing velocity, speed, acceleration, and position, we can use derivatives to analyze various types of populations, including those as diverse as bacteria colonies and cities. We can use a current population, together with a growth rate, to estimate the size of a population in the future. If [latex]P(t)[\/latex] is the number of entities present in a population, then the population growth rate of [latex]P(t)[\/latex] is defined to be [latex]P^{\\prime}(t)[\/latex].<\/p>\r\n<\/section>\r\n The population of a city is tripling every [latex]5[\/latex] years. If its current population is [latex]10,000[\/latex], what will be its approximate population [latex]2[\/latex] years from now?<\/p>\r\n [reveal-answer q=\"fs-id1169739299610\"]Show Solution[\/reveal-answer] Let [latex]P(t)[\/latex] be the population (in thousands) [latex]t[\/latex] years from now. Thus, we know that [latex]P(0)=10[\/latex] and based on the information, we anticipate [latex]P(5)=30[\/latex]. Now estimate [latex]P^{\\prime}(0)[\/latex], the current growth rate, using<\/p>\r\n By applying the average rate of change formula to [latex]P(t)[\/latex], we can estimate the population [latex]2[\/latex] years from now by writing<\/p>\r\n thus, in [latex]2[\/latex] years the population will be approximately [latex]18,000[\/latex].<\/p>\r\n Watch the following video to see the worked solution to this example.<\/p>\r\n
\r\n
\r\nThe population growth rate<\/strong> is the rate of change of a population and consequently can be represented by the derivative of the size of the population.<\/p>\r\npopulation growth rate<\/h3>\r\n
\r\n[hidden-answer a=\"fs-id1169739299610\"]<\/p>\r\n