Suppose a city's population, in thousands, t years after 1980 is represented by an exp. func. P(t). We know

Question

that in 1983 the city had a population of P2=310 thousand and in 1985 the city'd population was P5=220 thousand. Find the number (in thousands) by which the city's population decreased from the year 1984 to the year 1988. Round the answers to the nearest integer.

Answer 1

Hi,

If the population change is following an exponential function, you need to find the yearly percent change so you can write the exponential equation. To do this put two equations into your graphing calculator and find their intersection. Type the equations Y1 = 220 and y2 = 310(1-x)^2, The first equation represents 220,000 people in 1985 and the second equation represents 310,000 people in 1983, decreasing at the decimal rate 1 - x for 2 years. If you set your window so ymin=0, ymax = 310 and yscl = 100, you will see 2 intersections. The first one is the location you are interested in. Its x value is .157576, which means the population change from year to year is 1 - .157576 or .842424. Each year the population is84.2424% of the previous year. So your exponential function that fits this data is Y = 310(.842424)^x.

If you put this into your calculator. You will see when x = 0 that y = 310. That's your population in 1983, 0 years from when you first started measuring the population. Two years later in 1985, the population of 220,000 shows when x = 2. The year 1984 is 1 year after 1983 and 1988 is 5 years after 1983, so use x = 1 and x = 5 for the populations in those years. In 1984 the population was 261,151 and in 1988 the population was 131,527. So the difference in these numbers is 129,624.
That is how much the population decreased from '84 until '88.
This is clearly a buyer's market for houses! And we thought we had it bad!!

I hope this helps!! :-)

Answer 2

Make 1983 the base year. Then
P = 310,000a^(t - 1983)
a^2 = 22/31
Ln(a) = (1/2)Ln(22/31)
Ln(a) = -0.1714724
a = 0.8424235
310,000a - 310,000a^5 =
310,000(a - a^5) ≈ 129,625 ≈ 130,000