Question

The function f(x) = 2(3)^x represents the growth of a dragonfly population every year in a remote swamp. Rose wants to manipulate the formula to an equivalent form that calculates seven times a year, not just once a year. Which function is correct for Rose's purpose, and what is the new growth rate?

Answer 1

`f(x)=2(1.17)^(7x)` and growth rate is 17%.

Explanation:

The general exponential growth function is

`g(x)=a(1+r)^x` ...... (1)

where, a is initial value, r is growth rate and x is time.

The given function is

`f(x)=2(3)^(x)`

Rose wants to manipulate the formula to an equivalent form that calculates seven times a year, not just once a year.

Using the properties of exponents the given function can be written as

`f(x)=2(3^{(1)/(7)})^(7x)`

`f(x)=2(1.17)^(7x)`

It can be rewritten as

`f(x)=2(1+0.17)^(7x)` .... (2)

On comparing (1) and (2) we get

r=0.17

Therefore, the required function is
`f(x)=2(1.17)^(7x)` and growth rate is 17%.