Let's analyze the equation:
[tex]\[ P(t) = 30,000,000 \cdot \left(\frac{8}{9}\right)^t \][/tex]
Here, [tex]\( P(t) \)[/tex] represents the total number of plant species at time [tex]\( t \)[/tex] years since the ice age began. The initial number of plant species when [tex]\( t = 0 \)[/tex] is 30,000,000.
The term [tex]\(\left(\frac{8}{9}\right)^t\)[/tex] signifies how the number of plant species changes over time.
To understand how the number of plant species changes yearly, we need to examine the term [tex]\(\frac{8}{9}\)[/tex]:
1. Yearly Factor: The fraction [tex]\(\frac{8}{9}\)[/tex] indicates the rate at which the plant species change every year.
- Since [tex]\(\frac{8}{9}\)[/tex] is less than 1, it indicates a decrease or shrinking of the number of plant species.
2. Growth or Shrinkage:
- If the factor were greater than 1, it would indicate growth.
- If the factor is less than 1, it indicates shrinkage.
In this case, since [tex]\(\frac{8}{9} \approx 0.8889\)[/tex] is less than 1, we conclude that the number of plant species shrinks every year.
3. Factor of Change: The factor by which the number of plant species changes every year is exactly [tex]\(\frac{8}{9}\)[/tex].
Putting everything together:
- Every year, the number of plant species shrinks by a factor of [tex]\(\frac{8}{9}\)[/tex].
Thus, the completed sentence is:
Every year, the number of plant species shrinks by a factor of [tex]\(\frac{8}{9}\)[/tex].
