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Answer :
To solve the problem of identifying the exponential function that describes the decreasing size of a rainforest, we need to understand the behavior of exponential decay.
Given the situation:
- The rainforest is currently decreasing at a rate of 50% per year.
- The current size of the rainforest is 210,000 square miles.
- We need an exponential function to model the size of the rainforest after [tex]$t$[/tex] years.
We know that exponential decay can be modeled by the equation:
[tex]\[ F = \text{Initial amount} \times ( \text{Decay rate})^t \][/tex]
Here:
- The initial amount is 210,000 square miles.
- The decay rate is 0.5 (since the rainforest is decreasing at a rate of 50% each year, the remaining amount each year is 50% or 0.5 of the previous year).
So, our exponential decay function will be:
[tex]\[ F = 210,000 \times (0.5)^t \][/tex]
Looking at the options provided:
A. [tex]\( F = 210,000 (0.95)^t \)[/tex]
B. [tex]\( F = 220,000 (1.4)^t \)[/tex]
C. [tex]\( F = 210,000 (1.5)^t \)[/tex]
D. [tex]\( F = 210,000 (0.5)^t \)[/tex]
The correct answer is:
[tex]\[ \boxed{D. \, F = 210,000 (0.5)^t} \][/tex]
This function accurately represents the scenario where the size of the rainforest decreases by 50% per year starting from 210,000 square miles.
Given the situation:
- The rainforest is currently decreasing at a rate of 50% per year.
- The current size of the rainforest is 210,000 square miles.
- We need an exponential function to model the size of the rainforest after [tex]$t$[/tex] years.
We know that exponential decay can be modeled by the equation:
[tex]\[ F = \text{Initial amount} \times ( \text{Decay rate})^t \][/tex]
Here:
- The initial amount is 210,000 square miles.
- The decay rate is 0.5 (since the rainforest is decreasing at a rate of 50% each year, the remaining amount each year is 50% or 0.5 of the previous year).
So, our exponential decay function will be:
[tex]\[ F = 210,000 \times (0.5)^t \][/tex]
Looking at the options provided:
A. [tex]\( F = 210,000 (0.95)^t \)[/tex]
B. [tex]\( F = 220,000 (1.4)^t \)[/tex]
C. [tex]\( F = 210,000 (1.5)^t \)[/tex]
D. [tex]\( F = 210,000 (0.5)^t \)[/tex]
The correct answer is:
[tex]\[ \boxed{D. \, F = 210,000 (0.5)^t} \][/tex]
This function accurately represents the scenario where the size of the rainforest decreases by 50% per year starting from 210,000 square miles.
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