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The population of butterflies increases from 2000 to 2800 over 7 years. Which function best models the population [tex]x[/tex] years from the first measurement?

A. [tex]f(x)=2000(1+0.4)^{\frac{1}{7} x}[/tex]
B. [tex]f(x)=2000\left(1+0.4^7\right)^x[/tex]
C. [tex]f(x)=2000\left(1+\frac{0.4}{7}\right)^x[/tex]
D. [tex]f(x)=2000\left(1+\frac{0.4}{7}\right)^{7x}[/tex]



Answer :

GinnyAnswer
To determine which function best models the population of butterflies [tex]\( x \)[/tex] years from the first measurement, we start by considering the given conditions:

- Initial population ([tex]\( P_{0} \)[/tex]): 2000 butterflies
- Final population ([tex]\( P_{f} \)[/tex]): 2800 butterflies
- Time span ([tex]\( t \)[/tex]): 7 years

We need to evaluate each function and see which one produces values that reasonably match the population growth from 2000 to 2800 over the 7-year period.

### Option A: [tex]\( f(x) = 2000 \left( 1 + 0.4 \right)^{\frac{x}{7}} \)[/tex]

For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 2000 \left( 1 + 0.4 \right)^{\frac{1}{7}} \][/tex]
[tex]\[ f(1) \approx 2000 \times 1.049 \][/tex]
[tex]\[ f(1) \approx 2098.483 \][/tex]

### Option B: [tex]\( f(x) = 2000 \left( 1 + 0.4^7 \right)^x \)[/tex]

For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 2000 \left( 1 + 0.4^7 \right)^1 \][/tex]
[tex]\[ f(1) \approx 2000 \times 1.00164 \][/tex]
[tex]\[ f(1) \approx 2003.2768 \][/tex]

### Option C: [tex]\( f(x) = 2000 \left( 1 + \frac{0.4}{7} \right)^x \)[/tex]

For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 2000 \left( 1 + \frac{0.4}{7} \right)^1 \][/tex]
[tex]\[ f(1) \approx 2000 \times 1.05714 \][/tex]
[tex]\[ f(1) \approx 2114.286 \][/tex]

### Option D: [tex]\( f(x) = 2000 \left( 1 + \frac{0.4}{7} \right)^{7x} \)[/tex]

For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 2000 \left( 1 + \frac{0.4}{7} \right)^7 \][/tex]
[tex]\[ f(1) \approx 2000 \times 1.475 \][/tex]
[tex]\[ f(1) \approx 2950.977 \][/tex]

### Conclusion:

Given that the butterfly population grows from 2000 to 2800 over 7 years, we are looking for a model that captures a significant increase over time. From our evaluations:

- Option A: [tex]\( \approx 2098.483 \)[/tex]
- Option B: [tex]\( \approx 2003.2768 \)[/tex]
- Option C: [tex]\( \approx 2114.286 \)[/tex]
- Option D: [tex]\( \approx 2950.977 \)[/tex]

Model [tex]\( D \)[/tex] estimates a final population of approximately 2950.977, which is the closest to the actual final population of 2800. Therefore, the best function that models the population is:

[tex]\[ f(x) = 2000 \left( 1 + \frac{0.4}{7} \right)^{7x} \][/tex]
mathstudent55

Answer:

A.   [tex] f(x) = 2000(1 + 0.4)^{\frac{1}{7}x} [/tex]

Step-by-step explanation:

The number of years we know about is 7, so x = 7.

[tex] f(x) = 2000(1 + 0.4)^{\frac{1}{7}x} [/tex]

[tex] f(x) = 2000(1.4)^{\frac{1}{7} \times 7} [/tex]

[tex] f(x) = 2000(1.4)^{1} [/tex]

[tex] f(x) = 2000(1.4) [/tex]

[tex] f(x) = 2800 [/tex]

Answer: Choice A.

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