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The function below describes the population of caribou in a tundra, where [tex]f(t)[/tex] represents the number of caribou, in hundreds, and [tex]t[/tex] represents the time, in years.

[tex]f(t) = 1.8(1.2)^t[/tex]

Initially, the tundra has [tex]\(\square\)[/tex] [tex]\(\square\)[/tex] caribou, and every [tex]\(\square\)[/tex] , the number of caribou increases by a factor of 1.2.



Answer :

GinnyAnswer
Let's break down the given function [tex]\( f(t) = 1.8(1.2)^t \)[/tex] to understand its parts and answer the questions step-by-step:

1. Initial Population:
The term [tex]\( 1.8 \)[/tex] in the function [tex]\( f(t) \)[/tex] represents the initial number of caribou in hundreds because at [tex]\( t = 0 \)[/tex],
[tex]\[ f(0) = 1.8 \cdot (1.2)^0 = 1.8 \cdot 1 = 1.8 \text{ (hundreds of caribou)}. \][/tex]
Therefore, the initial number of caribou in the tundra is [tex]\( 1.8 \times 100 = 180 \)[/tex] caribou.

2. Time Unit for Population Increase:
The exponent [tex]\( t \)[/tex] represents the time in years. Hence, the population factor increase happens every 1 year.

3. Factor of Population Increase:
The base [tex]\( 1.2 \)[/tex] in the function indicates the population increases by a factor of [tex]\( 1.2 \)[/tex] each year.

By filling in these values in the context of the given function:

Initially, the tundra has [tex]\( \boxed{180} \)[/tex] caribou, and every [tex]\( \boxed{1 \text{ year}} \)[/tex], the number of caribou increases by a factor of [tex]\( \boxed{1.2} \)[/tex].