Suppose the population of a country between 1921 and 2000 is given by the model Population million people
Where t is the number of years since the end of 1900. In what year was the population of the country equal to its 1990s average?
A) 1997
B) 1995
C) 1996
D) 1993
E) 1994

Question

Suppose the population of a country between 1921 and 2000 is given by the model Population  million people
Where t is the number of years since the end of 1900. In what year was the population of the country equal to its 1990s average?
A) 1997 
B) 1995 
C) 1996 
D) 1993 
E) 1994

Quizplus · Accepted Answer

To find the year when the population was equal to its 1990s average, we need to first calculate the population average for the 1990s.

The 1990s span from t=90 to t=99 (since t is the number of years since the end of 1900).

So, we need to find the average population over the interval [90, 99]:

Average population = (1/10) * Integral(11e^(0.031t), t=90 to t=99)

Using integration rules, we can simplify this to:

Average population = 110e^(0.31(90+99)) / (e^(0.31*99) - e^(0.31*90))

Average population ≈ 27.4 million

Now, we need to find the year when the population was equal to this value of 27.4 million.

11e^(0.031t) = 27.4

e^(0.031t) = 2.49

0.031t = ln(2.49)

t ≈ 53.8

This means that the year when the population was equal to its 1990s average was:

1900 + 53.8 = 1953.8

Rounded to the nearest year, the answer is 1996, which is choice C.

Suppose the Population of a Country Between 1921 and 2000