Question

The function f(x) = 20(3)^x represents the growth of a fish population every year in a local lake. Jesse wants to manipulate the formula to an equivalent form that calculates every half-year, not every year. Which function is correct for Jesse's purposes?

Answer 1

The correct answer is:

`f(x) = 20(2)^x`

Step-by-step explanation:

This is an equation of the form

`f(x) = a* b^x`,

where a is the initial population, x is the amount of time, and b = 1+r, where r is the rate at which the population increases per year.

Matching this with our function, we see that a = 20 and b = 3; this means 1+r=3, so r = 2. This means the rate at which the population increases is 200% per year.

This means that every half year, the population would increase by 200/2 = 100%; this means r = 1. This gives us

`f(x) = a* b^x</p><p>\\</p><p>\\f(x) = a* (1+r)^x</p><p>\\</p><p>\\f(x)=20* (1+1)^x</p><p>\\</p><p>\\f(x) = 20* 2^x`

Answer 2

We need the exponents law


`a^(bx)=(a^b)^x`
So if
a=3
b=1/2

then

`3^(1/2*x)=(3^(1/2))^x`
where x represent every half a year.