Question

the population growth over time in tiny town is modeled by the function rule $p(t) = 1,500(1.25)^t$, where $t$ is the number of years after the year 2000.
in the year 2000, smallberg’s population was 1,000 and increased by 40% each year.
compare the population growth of the two villages.
decide if each statement is true or false.
smallberg had a larger initial population than tiny town.
true
false
the population in smallberg and tiny town both grow exponentially.
true
false
in the year 2003, tiny town had a larger population than smallberg.
true
false
tiny town had a larger average rate of change in population than smallberg between the years 2000 and 2002.
true
false

Explanation:

Step1: Check initial populations

Tiny Town initial population (2000): $P(0)=1500(1.25)^0=1500$
Smallberg initial population (2000): 1000

Step2: Verify growth model type

Tiny Town: $P(t)=1500(1.25)^t$ (exponential)
Smallberg: grows 40% yearly, so population is $1000(1.4)^t$ (exponential)

Step3: Calculate 2003 populations

Tiny Town ($t=3$): $1500(1.25)^3=1500\times1.953125=2929.6875$
Smallberg ($t=3$): $1000(1.4)^3=1000\times2.744=2744$

Step4: Calculate avg rate of change (2000-2002)

Tiny Town: $\frac{1500(1.25)^2 - 1500}{2}=\frac{1500(1.5625-1)}{2}=\frac{1500\times0.5625}{2}=421.875$
Smallberg: $\frac{1000(1.4)^2 - 1000}{2}=\frac{1000(1.96-1)}{2}=\frac{1000\times0.96}{2}=480$

Answer:

1. Smallberg had a larger initial population than Tiny Town.

○ False

1. The population in Smallberg and Tiny Town both grow exponentially.

○ True

1. In the year 2003, Tiny Town had a larger population than Smallberg.

○ True

1. Tiny Town had a larger average rate of change in population than Smallberg between the years 2000 and 2002.

○ False