In 1970, the total enrollment in public universities and colleges in a country was 5.1 million students. By 2004, enrollment had risen to 14.3 million students. Assume enrollment can be modeled by exponential growth.
(a) Estimate the total enrollments in 1980, 1990, and 2000. (Round your answers to one decimal place.)
1980___ million
1990___ million
2000___ million
(b) How many years after 1970 until the enrollment doubles from the 2004 figure? (Round your answer to one decimal place.)
____ years
(c) By what percent is the enrollment increasing each year? (Round your answer to two decimal places.)
____ %
To estimate the total enrollments in 1980, 1990, and 2000, we can use the formula for exponential growth:
A = P * (1 + r)^t
Where:
A = final amount
P = initial amount (in 1970)
r = annual growth rate (unknown)
t = time in years
Given:
Initial enrollment in 1970 = 5.1 million students
Final enrollment in 2004 = 14.3 million students
To find the annual growth rate (r), we can use the formula:
r = (A/P) ^ (1/t) - 1
For 1980:
t = 1980 - 1970 = 10 years
r = (14.3/5.1)^(1/34) -1 = 0.069
A (1980) = 5.1 * (1 + 0.069)^10 = 9.4 million
For 1990:
t = 1990 - 1970 = 20 years
r = (14.3/5.1)^(1/34) -1 = 0.069
A (1990) = 5.1 * (1 + 0.069)^20 = 16.6 million
For 2000:
t = 2000 - 1970 = 30 years
r = (14.3/5.1)^(1/34) -1 = 0.069
A (2000) = 5.1 * (1 + 0.069)^30 = 51.3 million
(a) Estimated total enrollments in 1980, 1990, and 2000:
1980: 9.4 million
1990: 16.6 million
2000: 51.3 million
For (b), to find the number of years until the enrollment doubles from the 2004 figure:
t = (log2(2A/P)) / log2(1+r)
Where:
A = final amount (2004 enrollment doubled)
P = initial amount (2004 enrollment)
A = 14.3 * 2 = 28.6 million
t = (log2(2 * 28.6/14.3)) / log2(1 + 0.069) = 9.8 years
(b) The number of years after 1970 until the enrollment doubles from the 2004 figure is approximately 9.8 years.
For (c), to find the percentage increase each year, we can use the formula:
Percentage increase = (r) * 100
Percentage increase = 0.069 * 100 = 6.9%
(c) The enrollment is increasing by approximately 6.9% each year.
To estimate the total enrollments in 1980, 1990, and 2000, we can use exponential growth formula:
A = P * (1 + r)^t
Where:
A = final amount
P = initial amount
r = growth rate
t = time in years
Given:
Initial amount (P) in 1970 = 5.1 million students
Final amount (A) in 2004 = 14.3 million students
To find the growth rate (r), we can use the following formula:
r = (A / P)^(1 / t) - 1
Let's calculate the values for each year.
(a) Estimate the total enrollments in 1980, 1990, and 2000:
For 1980:
t = 1980 - 1970 = 10 years
r = (14.3 / 5.1)^(1 / 10) - 1
Now, substitute the values into the exponential growth formula:
A = 5.1 * (1 + r)^10
Calculate A and round the answer to one decimal place.
For 1990:
t = 1990 - 1970 = 20 years
r = (14.3 / 5.1)^(1 / 20) - 1
A = 5.1 * (1 + r)^20
Calculate A and round the answer to one decimal place.
For 2000:
t = 2000 - 1970 = 30 years
r = (14.3 / 5.1)^(1 / 30) - 1
A = 5.1 * (1 + r)^30
Calculate A and round the answer to one decimal place.
(b) How many years after 1970 until the enrollment doubles from the 2004 figure:
To find the number of years, we need to calculate the time (t) when the final amount doubles the initial amount.
So, we have:
A = 2 * 14.3
P = 14.3
r = (2 * 14.3 / 5.1)^(1 / t) - 1
Solve for t using logarithms:
log(2 * 14.3 / 5.1) / log(1 + r) = t
Calculate t and round the answer to one decimal place.
(c) By what percent is the enrollment increasing each year:
To find the percent increase each year, we'll use the formula:
Percent Increase = r * 100
Calculate the percent increase and round the answer to two decimal places.
By following these steps, you should be able to find the estimated enrollments, the number of years to double the enrollment, and the percent increase.
Let the mathematical model be
f(x)=ax^b
where x=years past 1970.
f(x)=population in millions
Note: much better accuracy is obtained by shifting the origin to 1970, the closest possible.
Substituting given data for 1970:
5.1=ax^0=a
so
f(x)=5.1x^b
For 2004,
14.3=5.1(34^(b))
or
b=log(14.3/5.1)/log(34)
=.292374812
The complete model is then
f(x)=5.1x^0.292374812
Substitute x=(1980-1970)=10
to find the estimated population for 1980, etc.