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On the first day of winter, an entire field of trees starts losing its flowers. The number of locusts remaining alive in this population decreases rapidly due to the lack of flowers for them to eat.

The relationship between the elapsed time, [tex]t[/tex], in days, since the beginning of winter, and the total number of locusts, [tex]N(t)[/tex], is modeled by the following function:
[tex]
N(t) = 8950 \cdot (0.7)^{2t}
[/tex]

Complete the following sentence about the daily percent change of the locust population.
Round your answer to the nearest percent.

Every day, there is a [tex]\square \%[/tex] removal from [tex]\square[/tex] the locust population.



Answer :

GinnyAnswer
Let's analyze the given function [tex]\( N(t) = 8950 \cdot (0.7)^{2t} \)[/tex], which models the number of locusts in the field over time in days. The goal is to determine the daily percent change in the locust population.

1. Understand the base of the exponential function:
- Here, the base of the exponent is [tex]\( 0.7 \)[/tex].

2. Determine the rate of change every two days:
- The expression [tex]\( (0.7)^{2t} \)[/tex] tells us how the population changes every two days.

3. Determine the daily rate of change:
- Since [tex]\( (0.7)^{2t} \)[/tex] represents the change over two days, we need to find the daily change by taking the square root of 0.7's rate:
- The factor change over one day is [tex]\((0.7^2)^{1/2}\)[/tex].

4. Calculate the daily change factor:
- [tex]\( (0.7^2) = 0.49 \)[/tex].
- Taking the square root of 0.49 gives us the one-day change factor.

5. Daily percent decrease:
- To convert this factor to a percentage, we use [tex]\( 100 \times (1 - \sqrt{0.49}) \)[/tex].
- This value represents the daily percent removal.

6. Round to the nearest percent:
- We obtain a daily percent decrease of 30%.

Thus, the completed sentence is:
"Every day, there is a 30% removal from the locust population."