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 exponential growth model. The formula for exponential growth is:

P = P0 * (1 + r)^t

Where P is the final population, P0 is the initial population, r is the growth rate, and t is the time period.

Given that the total enrollment in 1970 was 5.1 million students, and in 2004 it was 14.3 million students, we can calculate the growth rate (r) using the formula:

r = (P / P0)^(1 / t) - 1

Where P is the final population, P0 is the initial population, and t is the time period.

Using this formula, we can now calculate the estimated total enrollment for each year:

(a)

For 1980:

P0 = 5.1 million
P = ?
t = 1980 - 1970 = 10 years

r = (14.3 / 5.1)^(1/34) - 1

Now, we can use the formula for exponential growth:

P = 5.1 * (1 + ((14.3 / 5.1)^(1/34) - 1))^10

Calculating this, we get P ≈ 11.7 million students for 1980.

For 1990:

P0 = 5.1 million
P = ?
t = 1990 - 1970 = 20 years

r = (14.3 / 5.1)^(1/34) - 1

P = 5.1 * (1 + ((14.3 / 5.1)^(1/34) - 1))^20

Calculating this, we get P ≈ 25.9 million students for 1990.

For 2000:

P0 = 5.1 million
P = ?
t = 2000 - 1970 = 30 years

r = (14.3 / 5.1)^(1/34) - 1

P = 5.1 * (1 + ((14.3 / 5.1)^(1/34) - 1))^30

Calculating this, we get P ≈ 51.9 million students for 2000.

Therefore, the estimated total enrollments in 1980, 1990, and 2000 are:
1980 ≈ 11.7 million
1990 ≈ 25.9 million
2000 ≈ 51.9 million

(b)

To find out how many years after 1970 it takes for the enrollment to double from the 2004 figure, we need to find the time period (t) for which

P = 2 * 14.3 = 28.6 million students

Using the formula for exponential growth, we can solve for t:

P0 = 5.1 million
P = 28.6 million
r = (14.3 / 5.1)^(1/34) - 1

P = 5.1 * (1 + r)^t

28.6 = 5.1 * (1 + r)^t

Simplifying the equation, we get:

(1 + r)^t = 28.6 / 5.1

Taking the logarithm of both sides, we have:

t * log(1 + r) = log(28.6 / 5.1)

Now we solve for t:

t = log(28.6 / 5.1) / log(1 + r)

Calculating this, we get t ≈ 18.4 years.

Therefore, it takes approximately 18.4 years for the enrollment to double from the 2004 figure.

(c)

To find out by what percent the enrollment is increasing each year, we can use the formula:

r = (1 + r) - 1

Using the growth rate we found earlier:

r = (14.3 / 5.1)^(1/34) - 1

Calculating this, we get r ≈ 0.047

To convert this to a percentage, we multiply by 100:

r_percentage = r * 100

Calculating this, we get r_percentage ≈ 4.7%

Therefore, the enrollment is increasing by approximately 4.7% each year.