The populations, P, of six towns with time t in years are given by:
I) P=1000(1.08)^t
II) P= 600(1.12)^t
III) P = 2500(0.9)^t
IV) P=1200(1.185)^t
V) P=800(0.78)^t
VI) P=2000(0.99)^t
a. Which towns are growing in size? Which are shrinking?
b. Which town is growing the fastest? What is the annual percent growth rate for that town?
c. Which town is shrinking the fastest? What is the annual percent "decay" rate for that town?
d. Which town has the largest initial population (at t=0)? Which town has the smallest?
growing: 1,2,4
fastest growth: 4 @ 18.5%
fastest shrink: 5 @ 22%
largest: 6
smallest: 2
a. To determine which towns are growing in size and which are shrinking, we need to analyze the growth/decay rates of the population models.
I) P = 1000(1.08)^t - Growing
II) P = 600(1.12)^t - Growing
III) P = 2500(0.9)^t - Shrinking
IV) P = 1200(1.185)^t - Growing
V) P = 800(0.78)^t - Shrinking
VI) P = 2000(0.99)^t - Shrinking
Therefore, towns I, II, and IV are growing in size, while towns III, V, and VI are shrinking.
b. To determine which town is growing the fastest, we can compare the growth rates of the models by looking at their base values (the number multiplied by the growth factor).
The growth percentage can be calculated as follows:
I) Growth percentage = (1.08 - 1) * 100 = 8%
II) Growth percentage = (1.12 - 1) * 100 = 12%
III) Decay percentage = (0.9 - 1) * 100 = -10%
IV) Growth percentage = (1.185 - 1) * 100 = 18.5%
V) Decay percentage = (0.78 - 1) * 100 = -22%
VI) Decay percentage = (0.99 - 1) * 100 = -1%
Town IV has the fastest growth rate, which is 18.5% annually.
c. To determine the town shrinking the fastest, we can compare the decay rates of the models by looking at their base values (the number multiplied by the decay factor).
Town V has the fastest decay rate, which is -22% annually.
d. To determine the town with the largest initial population (at t=0), we can simply look at the coefficients of the models.
Town III has the largest initial population with a size of 2500.
To determine the town with the smallest initial population, we look at the coefficients of the models.
Town II has the smallest initial population with a size of 600.
To determine which towns are growing in size and which are shrinking, we need to look at the exponential growth/decay model for each town. The general form of an exponential growth/decay model is given by P = P0 * r^t, where P is the population at time t, P0 is the initial population at t=0, and r is the growth/decay rate.
a. To determine whether a town is growing or shrinking, we can examine the value of the growth/decay rate (r) for each town. If r is greater than 1, the town is growing; if r is less than 1, the town is shrinking.
I) P = 1000(1.08)^t
The growth rate is 1.08, which is greater than 1. Therefore, town I is growing.
II) P = 600(1.12)^t
The growth rate is 1.12, which is greater than 1. Therefore, town II is growing.
III) P = 2500(0.9)^t
The growth rate is 0.9, which is less than 1. Therefore, town III is shrinking.
IV) P = 1200(1.185)^t
The growth rate is 1.185, which is greater than 1. Therefore, town IV is growing.
V) P = 800(0.78)^t
The growth rate is 0.78, which is less than 1. Therefore, town V is shrinking.
VI) P = 2000(0.99)^t
The growth rate is 0.99, which is less than 1. Therefore, town VI is shrinking.
b. To determine which town is growing the fastest, we need to compare the growth rates (r) of each town. The town with the highest growth rate is growing the fastest. Additionally, we can calculate the annual percent growth rate for that town by subtracting 1 from the growth rate and converting it to a percentage.
Comparing the growth rates:
I) growth rate = 1.08
II) growth rate = 1.12
IV) growth rate = 1.185
The town with the highest growth rate is town IV, which is growing the fastest. To calculate the annual percent growth rate, we subtract 1 from 1.185 and convert it to a percentage: (1.185 - 1) * 100% = 18.5% annual percent growth rate for town IV.
c. To determine which town is shrinking the fastest, we need to compare the decay rates (r) of each town. The town with the smallest decay rate is shrinking the fastest. Additionally, we can calculate the annual percent "decay" rate by subtracting the decay rate from 1 and converting it to a percentage.
Comparing the decay rates:
III) decay rate = 0.9
V) decay rate = 0.78
VI) decay rate = 0.99
The town with the smallest decay rate is town V, which is shrinking the fastest. To calculate the annual percent "decay" rate, we subtract the decay rate from 1 and convert it to a percentage: (1 - 0.78) * 100% = 22% annual percent "decay" rate for town V.
d. To determine the largest and smallest initial populations (at t=0), we need to find the initial population (P0) for each town.
I) P = 1000(1.08)^t
The initial population (P0) for town I is 1000.
II) P = 600(1.12)^t
The initial population (P0) for town II is 600.
III) P = 2500(0.9)^t
The initial population (P0) for town III is 2500.
IV) P = 1200(1.185)^t
The initial population (P0) for town IV is 1200.
V) P = 800(0.78)^t
The initial population (P0) for town V is 800.
VI) P = 2000(0.99)^t
The initial population (P0) for town VI is 2000.
The town with the largest initial population is town III with an initial population of 2500. The town with the smallest initial population is town II with an initial population of 600.