+4 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?
+128460 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
