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# word problem

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In 1969, in a report entitled Resources and Man, the U.S. National Academy of Sciences concluded that a world population of 10 billion “is close to (if not above) the maximum that an intensely managed world might hope to support with some degree of comfort and individual choice.” For this reason, 10 billion is called the carrying capacity of Earth. In 1999, the world population reached 6 billion and in 2011 the world population reached 7 billion. If exponential growth continues at this rate, in what year will Earth’s carrying capacity be reached?

Jan 31, 2021
edited by teloketo  Jan 31, 2021

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We  have  the exponential  function

P  =  ab^x       where  a  is  the population in 1999      P  is  the population in 2011   , x  is  the  number of years  between 1999 and  2011   =  12

We need to find "b"

So

7 =  6b^(12)      divide both sides by  6

(7/6)  = b^12       take the 12th root of both sides

b =   (7/6)^(1/12)

So we have

10  = 6 [(7/6) ^(1/12) ]^x           where  x  is the number of years after 1999

10 = 6 (7/6) ^( x/12)

divide both sides by  6

(10/6)  = (7/6)^(x/12)         take the log of both sides

log (10/6)  = log  (7/6) ^(x/12)       and  by a log property, we can write

log (10/6)  / log (7/6)   =  x /12

12 log (10/6)  / log (7/6)  = x  ≈  39.76  yrs      ≈  40 years after 1999   =   2039

Jan 31, 2021