In 1980, the population of the country Alford was 50,000 people and since was in
ID: 2912683 • Letter: I
Question
In 1980, the population of the country Alford was 50,000 people and since was increasing continuously by 5.5% per year for the next 30 years. On the other hand, the population of the country Olag was 60,000 and increasing by 3,000 people per year over the same time period.
A. For each country, write a formula expressing the population as a function of time t, where t is the number of years since 1980.
B. Approximately, what was the population of each country in 2008?
C. In how many years will the population of Alford reach approximately 75,000? (Use logs to solve)
D. In what year will the population of Alford surpass the population of Olag? (Solve graphically)
Explanation / Answer
Let t=0 denote the year 1980, A denotes Alford and O denote the country Olag.
t=0 P0(A) = 50,000 P0(O) = 60,000
A. Population as a function of time t
Alford: Population grows @ 5.5% per year implies by the end of each year population is 105.5% of previous one.
So, PA = (1.055)t P0(A), 0?t?30
Olag: Population grows by 3000 every year.
So, PO= P0(O) + 3000t 0?t?30
B. Population in 2008 (t=28)
PA = (1.055)t P0(A) = (1.055)28 * 50000 = 223,892
PO= P0(O) + 3000t = 60000 + 3000*28 = 144,000
C. T for population of Alford to reach 75,000
75000 = (1.055)t *50000
1.5 = (1.055)t
? t = ln(1.5)/ln(1.055) = 7.573 years
D. Year in which Alford population surpasses that of Olag
On plotting graph using a graphing calculator the intersection point refers to the year. It will be around 12.5 years.