Question

On November 27, 1993, the New York Times reported that wildlife biologists have found a direct link between the increase in the human population in Florida and the decline of the local black bear population. From 1953 to 1993, the human population increased, on average, at a rate of 8% per year, while the black bear population decreased at a rate of 6% per year. In 1953 the black bear population was 11,000.

(a) The 1993 human population of Florida was 13 million. What was the human population in 1953?
_______________people (Round to nearest whole person. Do not include commas in your answer.) help (numbers)

(b) Find the black bear population for 1993.
___________________bears (Round to nearest whole bear. Do not include commas in your answer.) help (numbers)

(c) Had this trend continued, when would the black bear population have numbered less than 90?
In the year ______________ help (numbers)

Answer 1

(a). Let the human population of Florida be x million in 1953. Since the human population increased, on average, at a rate of 8% per year between 1953 and, hence the human population of Florida in 1993 is x*(1.08)^40-1 = x* (1.08)³⁹( the nth term of a geometric series with 1^st term a and common ratio r is ar^n-1).

Hence, x* (1.08)³⁹=13 so that x = 13/ (1.08)³⁹ = 13/20.115297868 = 0.0.646274297 = 0.5 million ( on rounding off to 2 decimal places).

(b). In 1953,the black bear population was 11,000 and it decreased at a rate of 6% per year. Hence in 1993, the black bear population would be 11000(0.94)^40-1 = 11000(0.94)³⁹ = 11000*0.08953365 = 984.87 = 985 ( on rounding off to the nearest whole number).

(c). If the declining trend had continued, let the black bear population become 90, n years from 1953. Then 11000(0.94)^n-1 = 90 or, (0.94)^n-1 = 90/11000. Now, on taking log of both the sides, we get (n-1)log 0.94 = (log 90-log 11000)(as log ab=log a +log b, log a/b=log a-log b and log a^b = b log a) or,(n-1)= (log 90-log 11000)/ (log 0.94) or, n-1 = ( 1.954242509-4.041392685)/(-0.026872146) or, n-1 = 2.087150176/0.026872146 or, n-1 = 77.66965005 so that n =78.66965005 . Thus, if the declining trend had continued, the black bear population would become less than 90 in the 79^th year since 1953 i.e. in 2032.