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

Answer:

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

Step-by-step 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

$f(x)=2(3^{\frac{1}{7}})^{7x}$

$f(x)=2(1.17)^{7x}$

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%.