SOLUTION: The population of a city has risen and fallen over 20 years interval. Its population may be modeled by the following functions: P(x)= 12000+8000sin( 0.628x), where the domain is t
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Question 1161877: The population of a city has risen and fallen over 20 years interval. Its population may be modeled by the following functions: P(x)= 12000+8000sin( 0.628x), where the domain is the years since 1980, and the range is the population of the city. What is the largest and smallest population of the city may have?' also in which year is the largest and smallest population? Answer by greenestamps(13203) (Show Source):
The value of sin(X) varies between -1 and +1; so the value of 12000+800sin(X) varies between 12000-8000 = 4000 and 12000+8000 = 20000.
The period of sin(aX) is . So the period of this function is
The initial value -- when x is 0 and sin(aX) is 0, is the "central" value of 12000.
Like any sine function, a maximum value occurs 1/4 of the way through a cycle and a minimum value occurs 3/4 of the way through a cycle.
Since the period of this function is 10 years, maximum values of 20000 will occur 2.5 and 12.5 years after 1980, which means during 1982 and 1992. And minimum values will occur 7.5 and 17.5 years after 1980, which means during 1987 and 1997.
