the censue bureau for a large country has reported that the country is becoming
ID: 2859305 • Letter: T
Question
the censue bureau for a large country has reported that the country is becoming more diverse. The projected population (in millions) of a certain minority is modeled by the ezponential function h(t)=37.85(1.029)^t where t=o corresponds to 2000 and 0 less than or equal to t less than or equal to 50. a.
A.Estimate in what year the minority's population will double the 2005 population of 43.67 million. Use the algebra of logarithms to solve this problem.
The minority's population will double the 2005 population in ___. (Round down to the nearest whole number.)
B. The projected population (in millions) of another minority is modeled by the exponential function h left parenthesis t right parenthesis equals 11.26 left parenthesis 1.027 right parenthesis Superscript th(t)=11.26(1.027)t where tequals=0 corresponds to 2000 and 0less than or equalstless than or equals50. Estimate in what year this minority's population will double the 2005 population of 12.8612.86 million.
This minority's population will double the 2005 population in ___. (Round down to the nearest whole number.)
Explanation / Answer
A) 2h = 37.85(1.029)^t
=> 2(43.67) = 37.85(1.029)^t
=> 87.34/37.85 = (1.029)^t
=> 2.307530 = (1.029)^t
Taking natural log bith sides:
ln(2.307530) = tln(1.029)
=> t = 0.836178/0.02859 = 29 (approx)
So in 2029 the population will double the population in 2005
(B) h(t) = 11.26(1.027)^t
Now 2h = 11.26(1.027)^t
=> 2(12.86) = 11.26(1.027)^t
=> 25.72/11.26 = (1.027)^t
=> 2.28419 = (1.027)^t
Taking natural log both sides, we get:
ln(2.28419) = tln(1.027)
=> t = 0.82601/0.026642 = 31
So the population doubles in 2031
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