Question

The function f(x) = 4(2)x represents the growth of a butterfly population every year in a remote swamp. Jan wants to manipulate the formula to an equivalent form that calculates five times a year, not just once a year. Which function is correct for Jan's purpose, and what is the new growth rate?

f(x) = 4(1.15)x; growth rate is 5%
f(x) = 4(1.15)5x; growth rate is 15%
f(x) = 4(2)x; growth rate is 200%
f(x) = 4(2)x, growth rate is 5%

Answer 1

f(x) = 4(1.15)^(5x); growth rate is 15%

Explanation:

The function f(x) = 4(2)x represents the growth of a butterfly population every year in a remote swamp. Jan wants to manipulate the formula to an equivalent form that calculates five times a year, not just once a year. The function that is correct for Jan's purpose is f(x) = 4(1.15)^(5x) and the new growth rate is 15%.

1+r = 1.15

Therefore, the growth rate is 15%

Answer 2

f(x) = 4(1.15)^(5x); growth rate is 15%

Explanation:

If x still represents the number of years, but we want to see a growth factor that corresponds to 1/5 of a year, we can write the formula as ...

`f(x)=4\cdot\left(2^{(1)/(5)}\right)^(5x)\approx 4\cdot 1.15^(5x)`

The growth factor is 1+r = 1.15, so r = 0.15 = 15%, the growth rate in a 1/5-year interval.

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Comment on the question and answer

If Jan wants to compute the growth in 1/5-year intervals, it isn't clear why the desired formula uses an exponent of 5x, signifying that x is measured in years. We would expect the independent variable to be the number of 1/5-year intervals, rather than 1/5 that number.